# Radical Equations Circuit Answer Key

Radical Match Name____ANSWERS_____ Directions: MATCH each radical equation with its correct solution.

Solving Radical Equations To solve a radical equation—an equation that contains radicals or rational exponents—you need to eliminate the radicals or rational exponents and obtain a polynomial equation.

Step 1 Isolate the radical on one side of the equation, if necessary.
Step 2 Raise each side of the equation to the same exponent to eliminate the radical and obtain a linear, quadratic, or other polynomial equation.
Step 3 Solve the equation that results in step 2.
Step 4 Check your answers using the original equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Example 1 Solve 3x+1 −3 =7 for x.
Solution 3x+1 −3 = 7
3x+1 = 10
Isolate ()3x +1 2.
x + 1 = 100
3x = 99
x = 33

Solving with radicals and imaginary numbers (Unit 6 test) Solving for variables by getting rid of radicals.

Chapter 5 - Rational Exponents and Radicals
Chapter 6 - Quadratic Equations and Inequalities
Systems of Linear Equations

Solving Radical Equations SOLVING A RADICAL EQUATION
To solve a radical equation—an equation that contains radicals or rational exponents—you need to eliminate the radicals or rational exponents and obtain a polynomial equation.

The key to either technique is that students practice MP6 by ensuring that their answer in simplest radical form makes sense.

Solving Radical Equations and Inequalities Solve each equation.

Check your answers using the original equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Apply the rule xm n = n√xm x m n = x m n to rewrite the exponentiation as a radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides of the equation to the power of the index of the radical.

Solving Radical Equations and Inequalities Solve each equation.

Solving Radical Equations and Inequalities Solve radical equations by raising both sides


A nutritionist recommends that an adult male consumes 55 grams of fat per day, with an acceptable deviation of at most 25 grams. Let "x" be the daily fat intake in grams. The acceptable range of fat intake is represented by the absolute-value inequality where x is within 25 grams of 55.


The Maclaurin Series expansion of \(e^x\) is given by \(f(x) = e^x\), which can be expanded as:
\[f(x) = f(0) + f'(0)\frac{x}{1!} + f''(0)\frac{x^2}{2!} + f'''(0)\frac{x^3}{3!} + ...\]

Given \(f(x) = e^x\), we find:
- \(f(0) = e^0 = 1\)
- \(f'(0) = e^0 = 1\)
- \(f''(0) = e^0 = 1\)
- \(f'''(0) = e^0 = 1\), and so on.

Thus, the series expansion of \(e^x\) is:
\[e^x = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + ...\]
which can be represented as:
\[e^x = \sum_{n=0}^\infty \frac{x^n}{n!}\]

Considering the series \(\sum_{n=0}^\infty \frac{(\ln5)^n}{n!}\), by comparing it with the equation for \(e^x\), we find \(x = \ln5\). Therefore:
\[\sum_{n=0}^\infty \frac{(\ln5)^n}{n!} = e^{\ln5} = 5\]

This shows that the series \(\sum_{n=0}^\infty \frac{(\ln5)^n}{n!}\) converges to 5.


The number 243 has several factors, including 1, 3, 9, 27, 81, and 243. Additionally, -1, -3, -9, -27, -81, and -243 are also factors.

Some key calculations involving 243 include:
- 243 ÷ 6 = 40.5
- 121.5 x 2 = 243
- 243 ÷ 9 = 27
- 243 ÷ 5 = 48.6 (approximately)
- 38 goes into 243 approximately 6.39 times
- 8 goes into 243 30 times with a remainder of 3

The prime factor of 243 is 3, as 3 to the 5th power equals 243. The opposite of 243 is -243. Other notable calculations include:
- 486 - 243 = 243
- 243 = 3^5
- If you divide 243 by 32, you get approximately 7.59375.


## #006: Is the continuum hypothesis relevant in physics?

*Category: Mathematical foundations - Tags: Set theory, Topology*

The problem examines the relevance of the continuum hypothesis in physical theories. In the general theory, any space of experimentally distinguishable cases forms a second-countable \(T_0\) topological space. This leads to the mathematical question of whether a topological space exists with a cardinality greater than countable but less than that of the continuum.

The physical significance lies in the cardinality of a set of experimentally distinguishable cases being at most that of the continuum, as it must allow a \(T_0\) and second-countable topology. The standard formulation of set theory (ZFC) does not determine the existence of sets with cardinality between countable and continuum, introducing the separate axiom of the "continuum hypothesis" (CH).

Considering the mathematical premise, the question arises whether this choice is relevant for physics. For instance, the set of all possible space-time events may have a cardinality between countable and continuum. The first step involves understanding whether such a set can be given a suitable topology, where each open set corresponds to an experimentally verifiable statement, as done in the general theory.


scipy.signal.blackman
--------------------

The `scipy.signal.blackman` function returns a Blackman window, a taper formed by using the first three terms of a summation of cosines. It is designed to have close to minimal leakage possible and is close to optimal, only slightly worse than a Kaiser window.

### Parameters

* `M`: int, the number of points in the output window. If zero or less, an empty array is returned.
* `sym`: bool, optional, defaults to True. When True, generates a symmetric window for use in filter design. When False, generates a periodic window for use in spectral analysis.

### Returns

* `w`: ndarray, the window with the maximum value normalized to 1.

### Definition

The Blackman window is defined as:

\[w(n) = 0.42 - 0.5 \cos(2\pi n/M) + 0.08 \cos(4\pi n/M)\]

### Notes

The Blackman window is used as one of many windowing functions for smoothing values in signal processing literature. It is also known as an apodization or tapering function, and is considered a "near optimal" tapering function, almost as good as the Kaiser window.

### References

* Blackman, R.B. and Tukey, J.W., (1958) The measurement of power spectra, Dover Publications, New York.
* Oppenheim, A.V., and R.W. Schafer. Discrete-Time Signal Processing. Upper Saddle River, NJ: Prentice-Hall, 1999, pp. 468-471.

### Examples

To plot the window and its frequency response:

```python
>>> from scipy import signal
>>> from scipy.fftpack import fft, fftshift
>>> import matplotlib.pyplot as plt
>>> import numpy as np

>>> window = signal.blackman(51)
>>> plt.plot(window)
>>> plt.title("Blackman window")
>>> plt.ylabel("Amplitude")
>>> plt.xlabel("Sample")

>>> plt.figure()
>>> A = fft(window, 2048) / (len(window)/2.0)
>>> freq = np.linspace(-0.5, 0.5, len(A))
>>> response = 20 * np.log10(np.abs(fftshift(A / abs(A).max())))
>>> plt.plot(freq, response)
>>> plt.axis([-0.5, 0.5, -120, 0])
>>> plt.title("Frequency response of the Blackman window")
>>> plt.ylabel("Normalized magnitude [dB]")
>>> plt.xlabel("Normalized frequency [cycles per sample]")
```


## Exercise 1

### Functions Challenge 1 – Kilometers to miles converter

Write a function called **km_to_miles** that takes kilometers as a parameter, converts it into miles, and returns the result.

## 2

### Functions Challenge 2 – Divisible by 11 checker

Write a function called **is_divisible_by_11** that takes an integer as a parameter and returns whether it is divisible by 11 or not.

## 3

### Functions Challenge 3 – Highest number

Write a function called **get_highest** that takes 2 numbers as parameters and returns the highest of the 2 numbers.

## 4

### Functions Challenge 4 – Hexagon area calculator

Write a function called **hexagon_area** that takes the length of a side of a regular hexagon as a parameter and returns the area of the hexagon.

## 5

### Functions Challenge 5 – Sleeps until Christmas

Write a function called **sleeps_until_xmas** that calculates how many days it is until Christmas and returns the number of sleeps left until the big day.

## 6

### Functions Challenge 6 – Is a palindrome?

Create a function called **is_palindrome** that checks to see if a given string is a palindrome or not.

## 7

### Functions Challenge 7 – Fuel cost calculator

Write a function called **fuel_cost** that takes a distance as a required argument, mpg (default 50 mpg) and fuel cost (default $1 a liter) as optional arguments. The function should return the cost in dollars.

## 8

### Functions Challenge 8 – Most common character

Create a function called **most_common_char** that takes a string as an argument and returns the most common character in that string.

## 9

### Functions Challenge 9 – Is a prime number?

Write a function called **is_prime** that takes an integer as an argument and checks if it is a prime number or not, returning a Boolean.

## 10

### Functions Challenge 10 – All Prime numbers

Write a function called **primes** that takes an integer value as an argument and returns a list of all prime numbers up to that number.

## 11

### Functions Challenge 11 – Highest Common Factor

Write a function called **hcf** that takes 2 integer values and returns the highest common factor of the numbers.

## 12

### Functions Challenge 12 – Valid Date

Write a function called **is_valid_date** that takes a date in string format DD/MM/YYYY and checks to see if it is a valid date and in the correct format, considering leap years.

## 13

### Functions Challenge 13 – Postcode Checker

Write a function called **is_valid_postcode** that takes a post code as an argument and returns whether it is a valid post code, following the format: [2 letters] [1 or 2 Digits] [Space] [1 Digit] [2 letters].

## 14

### Functions Challenge 14 – Check Valid Email Address

Write a function called **is_valid_email** that takes an email address as an argument and returns True/False depending on whether it is a valid email address, following these rules:
- At least 1 character before the @ symbol
- Contains an @ symbol
- At least 1 character after the @ symbol and before the period
- At least 1 character after the last period
- Maximum 256 characters
- Starts with a letter or a number

## 15

### Functions Challenge 15 – Caesar Cipher Encryptor / Decryptor

Create a function called **encrypt** that takes some text (String) and a Shift (Integer) and then encrypts the text using the Caesar Cipher algorithm, returning the encrypted text. Create a second function to decrypt an encrypted string, using the same input parameters and returning the decrypted text.

## 16

### Functions Challenge 16 – Find all palindromes

Create a function called **get_palindromes** that is given a string and returns a list of all the palindromes in the string.

## 17

### Functions Challenge 17 – Sort Competitors List

Create a function that takes a 2D list of competitor names and league (e.g., Bob Jones, “Bronze”) as well as an optional parameter “Reverse”. The function should return the list sorted by league (Bronze, Silver, Gold, Ruby, Diamond), reversed if the optional reversed parameter is set to true.


## Understanding Centimeters and Inches

To convert centimeters to inches, it's essential to know the conversion factor. One centimeter is equal to 0.3937 inches.

## Definition: Centimeter

A centimeter is a base unit of length, equivalent to 0.01 meters. It is used in the CGS system, maps, home repair, and various areas of life.

## Definition of Inch

An inch is a unit of length in the UK and US customary systems of measurement, equal to 1/12 of a foot or 1/36 of a yard.

## Converting Centimeters to Inches

To convert centimeters to inches, multiply the value in centimeters by the conversion factor of 0.3937. For example, 1 cm is equal to 0.3937 inches.

## Converting 21.59 cm to Inches

Using the conversion factor, 21.59 cm can be converted to inches as follows:
Value in inches = value in cm × 0.3937
So, 21.59 cm to inches = 21.59 cm × 0.3937 = 8.499983 inches

## Conversion Table

The following table shows the conversion of different centimeter values to inches:

| cm | inches |
| --- | --- |
| 21.19 | 8.342503 |
| 21.24 | 8.362188 |
| 21.29 | 8.381873 |
| 21.34 | 8.401558 |
| 21.39 | 8.421243 |
| 21.44 | 8.440928 |
| 21.49 | 8.460613 |
| 21.54 | 8.480298 |
| 21.59 | 8.499983 |
| 21.64 | 8.519668 |
| 21.69 | 8.539353 |
| 21.74 | 8.559038 |
| 21.79 | 8.578723 |
| 21.84 | 8.598408 |
| 21.89 | 8.618093 |
| 21.94 | 8.637778 |

## Frequently Asked Questions

- What is 1 cm in inches?
  Answer: 0.3937 inches
- What is the conversion factor from cm to inches?
  Answer: 0.3937
- How many inches is equal to 1 cm?
  Answer: 0.3937 inches
- What is 1 cm equivalent to in inches?
  Answer: 0.3937 inches
- What is 21.59 cm in inches?
  Answer: 8.499983 inches
- How do you calculate cm to inches?
  Answer: Value in inches = value in cm × 0.3937


To convert whole number percentages using a calculator, you need to convert the interest rate to a fraction. For a loan with an interest rate of 12 and 1/4%, you calculate it as follows: 12 and 1/4 = [(4x12 + 1) / 4] X (1/100) = (49/4) X (1/100) = 49 / 400. So, for a $400 payment, $49 of the total $400 is interest.

Changing a fraction to a percent and a percent to a fraction are simple calculations. To change a fraction to a percent, divide the numerator by the denominator, then multiply the result by 100. The result is the fraction as a percent. Simple calculations can be done by hand and checked using a fraction to percent converter.

A common fraction is written as a number over another number with a line drawn between them. The top number is called the numerator, and the bottom is the denominator. In a calculator, it is possible to change a common fraction into its equivalent as a percentage by putting the numerator as the amount of change and the denominator as the total amount.

To calculate a percentage of a percentage, convert both percentages to fractions of 100 or to decimals and multiply them. For example, 50% of 40% is calculated as: 50 / 100 × 40 / 100 = 0.50 × 0.40 = 0.20 = 20 / 100 = 20%. It is not correct to divide by 100 and use the percent sign at the same time.

To calculate percentages, you need to calculate the fraction that each product category contributes to total sales. The sum of such fractions would equal 1, i.e., 100%. When sales from one category are divided by total sales, the result will be a value between 0 and 1. If this value is then multiplied by 100, the result can be presented in percent.


Q: Name all the factors for 255?

The factors of 255 are 1, 3, 5, 15, 17, 51, 85, and 255. The prime factors are 3, 5, and 17.

To find the factors of 255, we can start by dividing it by its prime factors. Since 5 is a prime factor, we can divide 255 by 5 to get 51. Then, we can divide 51 by 3 to get 17, which is also a prime factor.

The factors of 255 can be grouped into pairs: 1 and 255, 3 and 85, 5 and 51, and 15 and 17. The prime factorization of 255 is 3 x 5 x 17.

Common factors are factors that are common to two or more numbers. For example, the common factors of 136 and 255 are 1 and 17. The common factors of 170 and 255 are 1, 5, 17, and 85.

The greatest common factor (GCF) of 136, 204, and 255 is 17. To find the GCF, we need to list the factors of each number and find the largest factor that is common to all three numbers.

In contrast, the factors of 256 are 1, 2, 4, 8, 16, 32, 64, 128, and 256. The prime factorization of 256 is 2^8.

The factors of 170 are 1, 2, 5, 10, 17, 34, 85, and 170. The common factors of 170 and 255 are 1, 5, 17, and 85.

The prime factorization of 255 is 3 x 5 x 17. This means that 255 is a composite number because it has more factors than just 1 and itself.

The general factors of 510 are 1, 2, 3, 5, 6, 10, 15, 17, 30, 34, 51, 85, 102, 170, 255, and 510. The prime factors of 510 are 2, 3, 5, and 17.


A user wants to edit added calculated fields on a pivot table. They ask if it's possible to manually enter values to calculated fields on a pivot table or insert an editable column on a pivot table.

The expert responds that it's not possible to edit fields in a pivot table or add an editable column. The best option is to add another calculated field.

The user adds a calculated field with a formula set to equal 0, but wants to change the 0 to a different value in each cell. The expert explains that this can't be done directly in the pivot table, but the formula for the calculated field can be modified.

The user wants a formula that allows them to set individual cell values, but the expert explains that this can't be done with pivot tables. The user asks for an alternative and the expert suggests creating a custom column in the source data.

The user can create a custom column in the source data, but it won't be part of the pivot table. The expert recommends creating a custom column in the source data, which would be more flexible and dynamic.

The user is concerned that adding data beside the pivot table will offset values when the pivot table is refreshed. The expert agrees and suggests finding a way to create a custom column in the source data that won't disrupt the original data.

The expert recommends placing the custom column on the left-most part of the source data, hiding it, and showing it with the pivot table. This won't distort the visual aspect of the source data but will help with the customized data needed.

The user can insert a blank column on the left-most part of the source data, move the initial data to the next column, and enter custom data in the new column. In the pivot table, the custom data can be shown while hiding the original data.


The manner in which complex numbers represent 2D rotations is adapted to use quaternions for 3D rotations. Let H represent the set of quaternions, where each quaternion, h, is a four-dimensional vector represented as h = w + xi + yj + zk, with w, x, y, and z being real numbers. The symbols i, j, and k denote the "imaginary" components of the quaternion, with i^2 = j^2 = k^2 = -1, and ij = -ji = k, jk = -kj = i, and ki = -ik = j.

Quaternion multiplication can be derived from these relationships, resulting in a non-commutative operation. This means that the order of quaternions matters when multiplying them. The set of quaternions forms a group under this operation, but it is not commutative, which is also true of 3D rotations.

For convenience, quaternion multiplication can be expressed in terms of vector multiplications, dot products, and cross products. Let v be a three-dimensional vector representing the final three quaternion components. The first component of h is w, and the final three components are given by the vector v.

Unit quaternions, which satisfy the condition w^2 + x^2 + y^2 + z^2 = 1, are used to represent 3D rotations. The set of unit quaternions forms a subgroup because the multiplication of unit quaternions yields a unit quaternion, and the other group axioms hold.

A mapping from unit quaternions to SO(3) can be defined, where each unit quaternion h maps to a matrix. This matrix can be verified as orthogonal and has a determinant of 1, making it an element of SO(3). However, this representation is not unique, as the quaternions h and -h represent the same rotation.

To fix this, the identification trick can be used, where antipodal points on the unit sphere are identified. This means that h and -h are considered equivalent, resulting in a two-to-one mapping from unit quaternions to SO(3). The set of unit quaternions can be visualized as a sphere, where antipodal points are identified, resulting in a space that is homeomorphic to RP^3.

To force uniqueness of rotations, restrictions can be made to stay in the "upper half" of the sphere. For example, requiring w ≥ 0, with special handling for the boundary case where w = 0. If w = 0, then x ≥ 0, and if x = 0, then y ≥ 0. If y = 0, then only z ≥ 0 is allowed. These restrictions ensure that each rotation is represented by a unique quaternion, but it is essential to remember the connectivity of the space when applying these restrictions.


## Statement

The goal is to solve Einstein's riddle, a logical puzzle where `nbPeople` people have different names and characteristics. Each person can only have one characteristic from each category.

## Goal

Solve the riddle and print it as a grid, separated by space and newline, with each person in their own column and each characteristic category on its own line.

## Input

* Line 1: Two space-separated integers `nbCharacteristics` and `nbPeople` for the number of characteristics and the number of people.
* Next `nbCharacteristics` lines: The list of all characteristics in each characteristic category (one category per line), space-separated.
* Next line: `N`, the number of relational links.
* Next `N` lines: One relational link per line, with each link formatted as either `A & B` or `A ! B`.

## Output

* Line 1 to line `nbCharacteristics`: The grid of characteristics representing the solution to the riddle, line by line.

## Constraints

1 <= `nbCharacteristics`, `nbPeople` < 10.

## Example

Input:
4 4
Amelia Bob Charles Daniel
Autobus Car Bicycle Roller
Herb Bush Tree Flower
Turtle Ant Rhinoceros Elephant
15
Charles & Bicycle
Flower & Bob
Elephant ! Daniel
Elephant & Autobus
Rhinoceros & Bicycle
Amelia & Turtle
Tree & Turtle
Herb ! Car
Bush ! Daniel
Elephant & Bob
Tree & Car
Roller & Ant
Flower ! Daniel
Tree ! Elephant
Bob

Output:
Amelia Bob Charles Daniel
Car Autobus Bicycle Roller
Tree Flower Bush Herb
Turtle Elephant Rhinoceros Ant


Squares to Stairs is a math problem that involves finding the number of blocks in a staircase. The problem can be solved by recognizing a pattern in the number of blocks in each step. To introduce this task, ask students to think about how they see the shape growing. After students have an opportunity to draw and describe how they see the shape changing, they are ready to engage in group work and further study.

The problem can be done without relying on formal algebra. Start with a low number of stairs, like 2, and work out how many solutions there are. Then, continue doing that and notice the pattern in the numbers of ways for a particular number of stairs. The numbers of ways for a particular number of stairs is the sum of the numbers of ways to climb the stair for the previous two numbers of stairs.

This type of link is called a recurrence relationship. It is a great task that helps students understand the concept of recurrence and how to apply it to solve problems. The problem requires students to think critically and recognize patterns, making it an excellent example of a math problem that promotes critical thinking and problem-solving skills.

In addition to the recurrence relationship, the problem also involves the concept of perfect squares. A perfect square is an integer that is the square of an integer. For example, 9 is a perfect square because it is the square of 3. The problem requires students to understand the concept of perfect squares and how to apply it to solve the problem.

The solution to the problem involves finding the number of blocks in each step and recognizing the pattern in the number of blocks. The pattern can be used to find the number of blocks in any step, making it a useful tool for solving the problem. The problem is an example of a math problem that requires critical thinking and problem-solving skills, making it an excellent addition to any math curriculum.

The problem is also related to other math concepts, such as least squares and nonlinear least squares. These concepts involve finding the best fit line or curve to a set of data, and they are used in a variety of fields, including engineering, economics, and computer science. The problem provides a foundation for understanding these concepts and how to apply them to solve real-world problems.

In conclusion, the Squares to Stairs problem is a math problem that involves finding the number of blocks in a staircase. The problem can be solved by recognizing a pattern in the number of blocks in each step, and it requires critical thinking and problem-solving skills. The problem is related to other math concepts, such as perfect squares, least squares, and nonlinear least squares, making it an excellent addition to any math curriculum.