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Step 3.Simplify the expression. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Strategy to Solve Equations with Rational ExpressionsStep 1.Note any value of the variable that would make any denominator zero.Step 2.Find the least common denominator ofalldenominators in the equation.Step 3.Clear the fractions by multiplying both sides of the equation by the LCD.Step 4.Solve the resulting equation.S... | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Step 1.Note any value of the variable that would make any denominator zero. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Step 2.Find the least common denominator ofalldenominators in the equation. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Step 3.Clear the fractions by multiplying both sides of the equation by the LCD. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Step 4.Solve the resulting equation. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Step 5.Check. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
If any values found in Step 1 are algebraic solutions, discard them. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Check any remaining solutions in the original equation. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Property of Similar TrianglesIfÎABCÎABCis similar toÎXYZÎXYZ, then their corresponding angle measures are equal and their corresponding sides are in the same ratio. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
IfÎABCÎABCis similar toÎXYZÎXYZ, then their corresponding angle measures are equal and their corresponding sides are in the same ratio. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Problem Solving Strategy for Geometry ApplicationsStep 1.Readthe problem and make sure all the words and ideas are understood. Draw the figure and label it with the given information.Step 2.Identifywhat we are looking for.Step 3.Namewhat we are looking for by choosing a variable to represent it.Step 4.Translateinto an ... | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Step 1.Readthe problem and make sure all the words and ideas are understood. Draw the figure and label it with the given information. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Step 2.Identifywhat we are looking for. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Step 3.Namewhat we are looking for by choosing a variable to represent it. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Step 4.Translateinto an equation by writing the appropriate formula or model for the situation. Substitute in the given information. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Step 5.Solve the equationusing good algebra techniques. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Step 6.Checkthe answer in the problem and make sure it makes sense. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
Step 7.Answerthe question with a complete sentence. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-concepts |
complex rational expression : A complex rational expression is a rational expression in which the numerator or denominator contains a rational expression. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-terms |
extraneous solution to a rational equation : An extraneous solution to a rational equation is an algebraic solution that would cause any of the expressions in the original equation to be undefined. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-terms |
proportion : A proportion is an equation of the formab=cdab=cd, wherebâ0,dâ0.bâ0,dâ0.The proportion is read âaais tobb, asccis tod.d.â | https://openstax.org/books/elementary-algebra-2e/pages/8-key-terms |
rational equation : A rational equation is two rational expressions connected by an equal sign. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-terms |
rational expression : A rational expression is an expression of the formpqpq, wherepandqare polynomials andqâ0.qâ0. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-terms |
similar figures : Two figures are similar if the measures of their corresponding angles are equal and their corresponding sides are in the same ratio. | https://openstax.org/books/elementary-algebra-2e/pages/8-key-terms |
Note that the square root of a negative number is not a real number. | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
Every positive number has two square roots, one positive and one negative. The positive square root of a positive number is the principal square root. | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
We can estimate square roots using nearby perfect squares. | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
We can approximate square roots using a calculator. | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
When we use the radical sign to take the square root of a variable expression, we should specify thatxâ¥0xâ¥0to make sure we get the principal square root. | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
Simplified Square Rootaais considered simplified ifaahas no perfect-square factors. | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
Product Property of Square RootsIfa,bare non-negative real numbers, thenab=a·bab=a·b | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
Simplify a Square Root Using the Product PropertyTo simplify a square root using the Product Property:Step 1.Find the largest perfect square factor of the radicand. Rewrite the radicand as a product using the perfect square factor.Step 2.Use the product rule to rewrite the radical as the product of two radicals.Step 3.... | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
Step 1.Find the largest perfect square factor of the radicand. Rewrite the radicand as a product using the perfect square factor. | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
Step 2.Use the product rule to rewrite the radical as the product of two radicals. | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
Step 3.Simplify the square root of the perfect square. | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
Quotient Property of Square RootsIfa,bare non-negative real numbers andbâ0bâ0, thenab=abab=ab | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
Simplify a Square Root Using the Quotient PropertyTo simplify a square root using the Quotient Property:Step 1.Simplify the fraction in the radicand, if possible.Step 2.Use the Quotient Rule to rewrite the radical as the quotient of two radicals.Step 3.Simplify the radicals in the numerator and the denominator. | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
Step 1.Simplify the fraction in the radicand, if possible. | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
Step 2.Use the Quotient Rule to rewrite the radical as the quotient of two radicals. | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
Step 3.Simplify the radicals in the numerator and the denominator. | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
To add or subtract like square roots, add or subtract the coefficients and keep the like square root. | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
Sometimes when we have to add or subtract square roots that do not appear to have like radicals, we find like radicals after simplifying the square roots. | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
Product Property of Square RootsIfa,bare nonnegative real numbers, thenab=a·banda·b=abab=a·banda·b=ab | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
Special formulasfor multiplying binomials and conjugates:(a+b)2=a2+2ab+b2(aâb)(a+b)=a2âb2(aâb)2=a2â2ab+b2(a+b)2=a2+2ab+b2(aâb)(a+b)=a2âb2(aâb)2=a2â2ab+b2 | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
The FOIL method can be used to multiply binomials containing radicals. | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
Quotient Property of Square RootsIfa,bare non-negative real numbers andbâ0bâ0, thenab=abandab=abab=abandab=ab | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
Ifa,bare non-negative real numbers andbâ0bâ0, thenab=abandab=abab=abandab=ab | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
Simplified Square RootsA square root is considered simplified if there areno perfect square factors in the radicandno fractions in the radicandno square roots in the denominator of a fraction | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
no perfect square factors in the radicand | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
no fractions in the radicand | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
no square roots in the denominator of a fraction | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
To Solve a Radical Equation:Step 1.Isolate the radical on one side of the equation.Step 2.Square both sides of the equation.Step 3.Solve the new equation.Step 4.Check the answer. Some solutions obtained may not work in the original equation. | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
Step 1.Isolate the radical on one side of the equation. | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
Step 2.Square both sides of the equation. | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
Step 3.Solve the new equation. | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
Step 4.Check the answer. Some solutions obtained may not work in the original equation. | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
Solving Applications with FormulasStep 1.Readthe problem and make sure all the words and ideas are understood. When appropriate, draw a figure and label it with the given information.Step 2.Identifywhat we are looking for.Step 3.Namewhat we are looking for by choosing a variable to represent it.Step 4.Translateinto an ... | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
Step 1.Readthe problem and make sure all the words and ideas are understood. When appropriate, draw a figure and label it with the given information. | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
Step 2.Identifywhat we are looking for. | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
Step 3.Namewhat we are looking for by choosing a variable to represent it. | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
Step 4.Translateinto an equation by writing the appropriate formula or model for the situation. Substitute in the given information. | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
Step 5.Solve the equationusing good algebra techniques. | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
Step 6.Checkthe answer in the problem and make sure it makes sense. | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
Step 7.Answerthe question with a complete sentence. | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
Area of a Square | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
Falling ObjectsOn Earth, if an object is dropped from a height ofhhfeet, the time in seconds it will take to reach the ground is found by using the formulat=h4t=h4. | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
On Earth, if an object is dropped from a height ofhhfeet, the time in seconds it will take to reach the ground is found by using the formulat=h4t=h4. | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
Skid Marks and Speed of a CarIf the length of the skid marks isdfeet, then the speed,s, of the car before the brakes were applied can be found by using the formulas=24ds=24d. | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
If the length of the skid marks isdfeet, then the speed,s, of the car before the brakes were applied can be found by using the formulas=24ds=24d. | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
Properties of | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
ananwhennnis an even number andaâ¥0aâ¥0, thenananis a real numbera<0a<0, thenananis not a real numberWhennnis an odd number,ananis a real number for all values ofa.For any integernâ¥2nâ¥2, whennis oddann=aann=aFor any integernâ¥2nâ¥2, whennis evenann=|a|ann=|a| | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
aâ¥0aâ¥0, thenananis a real number | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
a<0a<0, thenananis not a real number | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
Whennnis an odd number,ananis a real number for all values ofa. | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
For any integernâ¥2nâ¥2, whennis oddann=aann=a | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
For any integernâ¥2nâ¥2, whennis evenann=|a|ann=|a| | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
ananis considered simplified ifahas no factors ofmnmn. | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
Product Property ofnth Rootsabn=an·bnandan·bn=abnabn=an·bnandan·bn=abn | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
Quotient Property ofnth Rootsabn=anbnandanbn=abnabn=anbnandanbn=abn | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
To combine like radicals, simply add or subtract the coefficients while keeping the radical the same. | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
Summary of Exponent Properties | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
Ifa,ba,bare real numbers andm,nm,nare rational numbers, thenProduct Propertyam·an=am+nam·an=am+nPower Property(am)n=am·n(am)n=am·nProduct to a Power(ab)m=ambm(ab)m=ambmQuotient Property:aman=amân,aâ0,m>naman=amân,aâ0,m>naman=1anâm,aâ0,n>maman=1anâm,aâ0,n>mZero Exponent Definitiona0=1a0=1,aâ0aâ0Quotien... | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
Product Propertyam·an=am+nam·an=am+n | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
Power Property(am)n=am·n(am)n=am·n | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
Product to a Power(ab)m=ambm(ab)m=ambm | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
Quotient Property:aman=amân,aâ0,m>naman=amân,aâ0,m>naman=1anâm,aâ0,n>maman=1anâm,aâ0,n>m | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
Zero Exponent Definitiona0=1a0=1,aâ0aâ0 | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
Quotient to a Power Property(ab)m=ambm,bâ0(ab)m=ambm,bâ0 | https://openstax.org/books/elementary-algebra-2e/pages/9-key-concepts |
index : anannis called theindexof the radical. | https://openstax.org/books/elementary-algebra-2e/pages/9-key-terms |
like radicals : Radicals with the same index and same radicand are called like radicals. | https://openstax.org/books/elementary-algebra-2e/pages/9-key-terms |
like square roots : Square roots with the same radicand are called like square roots. | https://openstax.org/books/elementary-algebra-2e/pages/9-key-terms |
nth root of a number : Ifbn=abn=a, thenbbis annth root ofaa. | https://openstax.org/books/elementary-algebra-2e/pages/9-key-terms |
principalnth root : The principalnth root ofaais writtenanan. | https://openstax.org/books/elementary-algebra-2e/pages/9-key-terms |
radical equation : An equation in which the variable is in the radicand of a square root is called a radical equation | https://openstax.org/books/elementary-algebra-2e/pages/9-key-terms |
rational exponents : Ifananis a real number andnâ¥2nâ¥2,a1n=ana1n=an.For any positive integersmandn,amn=(an)mamn=(an)mandamn=amnamn=amn. | https://openstax.org/books/elementary-algebra-2e/pages/9-key-terms |
rationalizing the denominator : The process of converting a fraction with a radical in the denominator to an equivalent fraction whose denominator is an integer is called rationalizing the denominator. | https://openstax.org/books/elementary-algebra-2e/pages/9-key-terms |
square of a number : Ifn2=mn2=m, thenmmis the square ofnn | https://openstax.org/books/elementary-algebra-2e/pages/9-key-terms |
square root notation : Ifm=n2m=n2, thenm=nm=n. We readmmas âthe square root ofmm.â | https://openstax.org/books/elementary-algebra-2e/pages/9-key-terms |
square root of a number : Ifn2=mn2=m, thennnis a square root ofmm | https://openstax.org/books/elementary-algebra-2e/pages/9-key-terms |
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