prompt stringlengths 32 115 | completion stringlengths 167 566 |
|---|---|
Find the roots of the quadratic equation: 7x^2 + 6x + 4 = 0 | 1. Identify coefficients: a=7, b=6, c=4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 6^2 - 4(7)(4) = 36 - 112 = -76.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-6 ± √76i) / 14.
Final Answer: x = -0.42857142857142855 ± 0.6226998490772392i |
Multiply the following 2x2 matrices:
A =
[-2, 5]
[-5, -3]
B =
[2, -1]
[1, 3] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-2*2) + (5*1) = 1
C12 = (A11 * B12) + (A12 * B22) = (-2*-1) + (5*3) = 17
C21 = (A21 * B11) + (A22 * B21) = (-5*2) + (-3*1) = -13
C22 = (A21 * B12) + (A22 * B22) = (-5*-1) + (-3*3) = -4
Resul... |
Multiply the following 2x2 matrices:
A =
[-1, 0]
[-3, 1]
B =
[0, -2]
[4, 2] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-1*0) + (0*4) = 0
C12 = (A11 * B12) + (A12 * B22) = (-1*-2) + (0*2) = 2
C21 = (A21 * B11) + (A22 * B21) = (-3*0) + (1*4) = 4
C22 = (A21 * B12) + (A22 * B22) = (-3*-2) + (1*2) = 8
Result Matr... |
In a right-angled triangle, the lengths of the two legs are a = 7 and b = 6. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 7^2 + 6^2 = c^2
2. Calculate the squares: 49 + 36 = c^2
3. Sum the squares: 85 = c^2
4. Take the square root of both sides: c = √85
Final Answer:... |
Compute the quotient and remainder of 73501 divided by 40. Show the long division steps. | To divide 73501 by 40:
1. Determine how many times 40 goes into the leading digits of 73501.
2. Multiply the divisor by this estimate, subtract from the current dividend portion, and bring down the next digit.
3. Repeat until all digits are processed.
Result: 73501 / 40 = 1837 with a remainder of 21.
Proof: (1837 * 40... |
Find the derivative of f(x) = (4x + 2)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 4x + 2.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 2. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 2)^2 * 4 = 12(4x ... |
Evaluate the logarithm: log_4(1024) | To evaluate log_4(1024), we ask the question: '4 raised to what power equals 1024?'
Let x be the unknown power: 4^x = 1024
Since 4^5 = 1024, it follows that x = 5.
Final Answer: 5 |
Find the derivative of f(x) = (5x + 5)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 5x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 5x + 5. Its derivative is h'(x) = 5.
4. Multiply g'(u) by h'(x): f'(x) = 3(5x + 5)^2 * 5 = 15(5x ... |
Compute the indefinite integral of f(x) = 3x^2. | To find the indefinite integral ∫ 3x^2 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 2 + 1 = 3.
2. Divide the coefficient by the new exponent: 3 / 3 = 1.
3. Combine and add the constant of integration C.
Final Answer: ∫ 3x^2 dx = 1x^3 + C |
Compute the quotient and remainder of 24800 divided by 98. Show the long division steps. | To divide 24800 by 98:
1. Determine how many times 98 goes into the leading digits of 24800.
2. Multiply the divisor by this estimate, subtract from the current dividend portion, and bring down the next digit.
3. Repeat until all digits are processed.
Result: 24800 / 98 = 253 with a remainder of 6.
Proof: (253 * 98) +... |
Multiply the following 2x2 matrices:
A =
[1, -2]
[2, 2]
B =
[-4, 0]
[-5, -4] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (1*-4) + (-2*-5) = 6
C12 = (A11 * B12) + (A12 * B22) = (1*0) + (-2*-4) = 8
C21 = (A21 * B11) + (A22 * B21) = (2*-4) + (2*-5) = -18
C22 = (A21 * B12) + (A22 * B22) = (2*0) + (2*-4) = -8
Result... |
Evaluate the logarithm: log_4(1024) | To evaluate log_4(1024), we ask the question: '4 raised to what power equals 1024?'
Let x be the unknown power: 4^x = 1024
Since 4^5 = 1024, it follows that x = 5.
Final Answer: 5 |
Compute the quotient and remainder of 45439 divided by 52. Show the long division steps. | To divide 45439 by 52:
1. Determine how many times 52 goes into the leading digits of 45439.
2. Multiply the divisor by this estimate, subtract from the current dividend portion, and bring down the next digit.
3. Repeat until all digits are processed.
Result: 45439 / 52 = 873 with a remainder of 43.
Proof: (873 * 52) ... |
Find the derivative of f(x) = (4x + 4)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 4x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 4. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 4)^2 * 4 = 12(4x ... |
Evaluate sin(30°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(30°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.5000 (exact form depends on root simplification if applicable) |
Find the roots of the quadratic equation: 6x^2 + -5x + -4 = 0 | 1. Identify coefficients: a=6, b=-5, c=-4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 25 - -96 = 121.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (5 ± √121) / 12.
5. Calculate roots: x1 = 1.3333333333333333, x2 = -0.5.
Final Answer: x = 1.3... |
Compute the indefinite integral of f(x) = 8x^3. | To find the indefinite integral ∫ 8x^3 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 3 + 1 = 4.
2. Divide the coefficient by the new exponent: 8 / 4 = 2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 8x^3 dx = 2x^4 + C |
Add the fractions 15/3 and 5/5. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 3 and 5, which is 15.
2. Convert 15/3 by multiplying top and bottom by 5: 75/15.
3. Convert 5/5 by multiplying top and bottom by 3: 15/15.
4. Add the numerators: 75 + 15 = 90.
5. The resulting fraction is 90/15.
6. Simplify by di... |
Multiply the following 2x2 matrices:
A =
[-3, -5]
[0, 3]
B =
[-5, 0]
[0, 2] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-3*-5) + (-5*0) = 15
C12 = (A11 * B12) + (A12 * B22) = (-3*0) + (-5*2) = -10
C21 = (A21 * B11) + (A22 * B21) = (0*-5) + (3*0) = 0
C22 = (A21 * B12) + (A22 * B22) = (0*0) + (3*2) = 6
Result M... |
Find the roots of the quadratic equation: 2x^2 + 8x + -9 = 0 | 1. Identify coefficients: a=2, b=8, c=-9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 64 - -72 = 136.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-8 ± √136) / 4.
5. Calculate roots: x1 = 0.9154759474226504, x2 = -4.91547594742265.
Final Ans... |
Add the fractions 14/4 and 7/10. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 4 and 10, which is 20.
2. Convert 14/4 by multiplying top and bottom by 5: 70/20.
3. Convert 7/10 by multiplying top and bottom by 2: 14/20.
4. Add the numerators: 70 + 14 = 84.
5. The resulting fraction is 84/20.
6. Simplify by ... |
Add the fractions 11/2 and 6/3. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 2 and 3, which is 6.
2. Convert 11/2 by multiplying top and bottom by 3: 33/6.
3. Convert 6/3 by multiplying top and bottom by 2: 12/6.
4. Add the numerators: 33 + 12 = 45.
5. The resulting fraction is 45/6.
6. Simplify by dividi... |
Compute the indefinite integral of f(x) = 5x^3. | To find the indefinite integral ∫ 5x^3 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 3 + 1 = 4.
2. Divide the coefficient by the new exponent: 5 / 4 = 5/4.
3. Combine and add the constant of integration C.
Final Answer: ∫ 5x^3 dx = 5/4x^4 + C |
Add the fractions 7/3 and 5/5. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 3 and 5, which is 15.
2. Convert 7/3 by multiplying top and bottom by 5: 35/15.
3. Convert 5/5 by multiplying top and bottom by 3: 15/15.
4. Add the numerators: 35 + 15 = 50.
5. The resulting fraction is 50/15.
6. Simplify by div... |
Find the roots of the quadratic equation: 7x^2 + -5x + 1 = 0 | 1. Identify coefficients: a=7, b=-5, c=1.
2. Calculate the discriminant: Δ = b^2 - 4ac = -5^2 - 4(7)(1) = 25 - 28 = -3.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (5 ± √3i) / 14.
Final Answer: x = 0.35714285714285715 ± 0.12371791482634838i |
Compute the indefinite integral of f(x) = 6x^2. | To find the indefinite integral ∫ 6x^2 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 2 + 1 = 3.
2. Divide the coefficient by the new exponent: 6 / 3 = 2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 6x^2 dx = 2x^3 + C |
Find the derivative of f(x) = (5x + 4)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 5x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 5x + 4. Its derivative is h'(x) = 5.
4. Multiply g'(u) by h'(x): f'(x) = 3(5x + 4)^2 * 5 = 15(5x ... |
In a right-angled triangle, the lengths of the two legs are a = 15 and b = 18. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 15^2 + 18^2 = c^2
2. Calculate the squares: 225 + 324 = c^2
3. Sum the squares: 549 = c^2
4. Take the square root of both sides: c = √549
Final A... |
Add the fractions 14/13 and 14/6. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 13 and 6, which is 78.
2. Convert 14/13 by multiplying top and bottom by 6: 84/78.
3. Convert 14/6 by multiplying top and bottom by 13: 182/78.
4. Add the numerators: 84 + 182 = 266.
5. The resulting fraction is 266/78.
6. Simpli... |
Find the derivative of f(x) = (5x + 3)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 5x + 3.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 5x + 3. Its derivative is h'(x) = 5.
4. Multiply g'(u) by h'(x): f'(x) = 3(5x + 3)^2 * 5 = 15(5x ... |
Multiply the following 2x2 matrices:
A =
[-4, -5]
[0, 0]
B =
[-4, -4]
[1, 1] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-4*-4) + (-5*1) = 11
C12 = (A11 * B12) + (A12 * B22) = (-4*-4) + (-5*1) = 11
C21 = (A21 * B11) + (A22 * B21) = (0*-4) + (0*1) = 0
C22 = (A21 * B12) + (A22 * B22) = (0*-4) + (0*1) = 0
Result ... |
Evaluate cos(45°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(45°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.7071 (exact form depends on root simplification if applicable) |
Find the roots of the quadratic equation: 5x^2 + -8x + -7 = 0 | 1. Identify coefficients: a=5, b=-8, c=-7.
2. Calculate the discriminant: Δ = b^2 - 4ac = 64 - -140 = 204.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (8 ± √204) / 10.
5. Calculate roots: x1 = 2.22828568570857, x2 = -0.6282856857085701.
Final A... |
Find the roots of the quadratic equation: 9x^2 + -7x + -7 = 0 | 1. Identify coefficients: a=9, b=-7, c=-7.
2. Calculate the discriminant: Δ = b^2 - 4ac = 49 - -252 = 301.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (7 ± √301) / 18.
5. Calculate roots: x1 = 1.3527417540498596, x2 = -0.5749639762720817.
Final... |
Evaluate the logarithm: log_2(16) | To evaluate log_2(16), we ask the question: '2 raised to what power equals 16?'
Let x be the unknown power: 2^x = 16
Since 2^4 = 16, it follows that x = 4.
Final Answer: 4 |
Find the roots of the quadratic equation: 9x^2 + 2x + -8 = 0 | 1. Identify coefficients: a=9, b=2, c=-8.
2. Calculate the discriminant: Δ = b^2 - 4ac = 4 - -288 = 292.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-2 ± √292) / 18.
5. Calculate roots: x1 = 0.8382226383686144, x2 = -1.0604448605908368.
Final ... |
In a right-angled triangle, the lengths of the two legs are a = 16 and b = 10. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 16^2 + 10^2 = c^2
2. Calculate the squares: 256 + 100 = c^2
3. Sum the squares: 356 = c^2
4. Take the square root of both sides: c = √356
Final A... |
In a right-angled triangle, the lengths of the two legs are a = 4 and b = 14. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 4^2 + 14^2 = c^2
2. Calculate the squares: 16 + 196 = c^2
3. Sum the squares: 212 = c^2
4. Take the square root of both sides: c = √212
Final Ans... |
Calculate the mean, population variance, and standard deviation for the dataset: [16, 15, 8, 12, 4] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (16 + 15 + 8 + 12 + 4) / 5 = 55 / 5 = 11.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [5.0, 4.0, -3.0, 1.0, -7.0]
Squared Differences: [25.0, 16.0, 9.0, 1.0, 49.0]
Sum... |
Evaluate the logarithm: log_4(256) | To evaluate log_4(256), we ask the question: '4 raised to what power equals 256?'
Let x be the unknown power: 4^x = 256
Since 4^4 = 256, it follows that x = 4.
Final Answer: 4 |
Multiply the following 2x2 matrices:
A =
[-1, 1]
[-5, 0]
B =
[4, -4]
[-4, -5] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-1*4) + (1*-4) = -8
C12 = (A11 * B12) + (A12 * B22) = (-1*-4) + (1*-5) = -1
C21 = (A21 * B11) + (A22 * B21) = (-5*4) + (0*-4) = -20
C22 = (A21 * B12) + (A22 * B22) = (-5*-4) + (0*-5) = 20
Re... |
Multiply the following 2x2 matrices:
A =
[-1, 3]
[-2, 2]
B =
[0, 1]
[3, 2] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-1*0) + (3*3) = 9
C12 = (A11 * B12) + (A12 * B22) = (-1*1) + (3*2) = 5
C21 = (A21 * B11) + (A22 * B21) = (-2*0) + (2*3) = 6
C22 = (A21 * B12) + (A22 * B22) = (-2*1) + (2*2) = 2
Result Matrix... |
Evaluate sin(45°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(45°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.7071 (exact form depends on root simplification if applicable) |
Find the derivative of f(x) = (4x + 3)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 4x + 3.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 3. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 3)^2 * 4 = 12(4x ... |
Evaluate tan(45°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(45°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 1.0000 (exact form depends on root simplification if applicable) |
In a right-angled triangle, the lengths of the two legs are a = 8 and b = 19. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 8^2 + 19^2 = c^2
2. Calculate the squares: 64 + 361 = c^2
3. Sum the squares: 425 = c^2
4. Take the square root of both sides: c = √425
Final Ans... |
Multiply the following 2x2 matrices:
A =
[3, -2]
[-2, 4]
B =
[3, 4]
[-2, 3] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (3*3) + (-2*-2) = 13
C12 = (A11 * B12) + (A12 * B22) = (3*4) + (-2*3) = 6
C21 = (A21 * B11) + (A22 * B21) = (-2*3) + (4*-2) = -14
C22 = (A21 * B12) + (A22 * B22) = (-2*4) + (4*3) = 4
Result M... |
Add the fractions 15/3 and 14/8. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 3 and 8, which is 24.
2. Convert 15/3 by multiplying top and bottom by 8: 120/24.
3. Convert 14/8 by multiplying top and bottom by 3: 42/24.
4. Add the numerators: 120 + 42 = 162.
5. The resulting fraction is 162/24.
6. Simplify ... |
Compute the indefinite integral of f(x) = 4x^4. | To find the indefinite integral ∫ 4x^4 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 4 + 1 = 5.
2. Divide the coefficient by the new exponent: 4 / 5 = 4/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 4x^4 dx = 4/5x^5 + C |
Evaluate the logarithm: log_3(243) | To evaluate log_3(243), we ask the question: '3 raised to what power equals 243?'
Let x be the unknown power: 3^x = 243
Since 3^5 = 243, it follows that x = 5.
Final Answer: 5 |
In a right-angled triangle, the lengths of the two legs are a = 12 and b = 4. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 12^2 + 4^2 = c^2
2. Calculate the squares: 144 + 16 = c^2
3. Sum the squares: 160 = c^2
4. Take the square root of both sides: c = √160
Final Ans... |
Evaluate tan(0°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(0°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.0000 (exact form depends on root simplification if applicable) |
Find the roots of the quadratic equation: 8x^2 + 8x + 9 = 0 | 1. Identify coefficients: a=8, b=8, c=9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 8^2 - 4(8)(9) = 64 - 288 = -224.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-8 ± √224i) / 16.
Final Answer: x = -0.5 ± 0.9354143466934853i |
Compute the indefinite integral of f(x) = 10x^5. | To find the indefinite integral ∫ 10x^5 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 5 + 1 = 6.
2. Divide the coefficient by the new exponent: 10 / 6 = 5/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 10x^5 dx = 5/3x^6 + C |
Compute the indefinite integral of f(x) = 3x^5. | To find the indefinite integral ∫ 3x^5 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 5 + 1 = 6.
2. Divide the coefficient by the new exponent: 3 / 6 = 1/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 3x^5 dx = 1/2x^6 + C |
Add the fractions 14/14 and 9/15. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 14 and 15, which is 210.
2. Convert 14/14 by multiplying top and bottom by 15: 210/210.
3. Convert 9/15 by multiplying top and bottom by 14: 126/210.
4. Add the numerators: 210 + 126 = 336.
5. The resulting fraction is 336/210.
6... |
Compute the quotient and remainder of 65625 divided by 71. Show the long division steps. | To divide 65625 by 71:
1. Determine how many times 71 goes into the leading digits of 65625.
2. Multiply the divisor by this estimate, subtract from the current dividend portion, and bring down the next digit.
3. Repeat until all digits are processed.
Result: 65625 / 71 = 924 with a remainder of 21.
Proof: (924 * 71) ... |
Find the roots of the quadratic equation: 10x^2 + 3x + -2 = 0 | 1. Identify coefficients: a=10, b=3, c=-2.
2. Calculate the discriminant: Δ = b^2 - 4ac = 9 - -80 = 89.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-3 ± √89) / 20.
5. Calculate roots: x1 = 0.32169905660283016, x2 = -0.6216990566028302.
Final A... |
Find the roots of the quadratic equation: 4x^2 + -8x + 6 = 0 | 1. Identify coefficients: a=4, b=-8, c=6.
2. Calculate the discriminant: Δ = b^2 - 4ac = -8^2 - 4(4)(6) = 64 - 96 = -32.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (8 ± √32i) / 8.
Final Answer: x = 1.0 ± 0.7071067811865476i |
Evaluate the logarithm: log_5(625) | To evaluate log_5(625), we ask the question: '5 raised to what power equals 625?'
Let x be the unknown power: 5^x = 625
Since 5^4 = 625, it follows that x = 4.
Final Answer: 4 |
Find the derivative of f(x) = (5x + 5)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 5x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 5x + 5. Its derivative is h'(x) = 5.
4. Multiply g'(u) by h'(x): f'(x) = 3(5x + 5)^2 * 5 = 15(5x ... |
Find the roots of the quadratic equation: 9x^2 + 3x + 2 = 0 | 1. Identify coefficients: a=9, b=3, c=2.
2. Calculate the discriminant: Δ = b^2 - 4ac = 3^2 - 4(9)(2) = 9 - 72 = -63.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-3 ± √63i) / 18.
Final Answer: x = -0.16666666666666666 ± 0.44095855184409843i |
Evaluate tan(60°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(60°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 1.7321 (exact form depends on root simplification if applicable) |
Find the derivative of f(x) = (4x + 3)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 4x + 3.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 3. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 3)^2 * 4 = 12(4x ... |
Compute the indefinite integral of f(x) = 6x^4. | To find the indefinite integral ∫ 6x^4 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 4 + 1 = 5.
2. Divide the coefficient by the new exponent: 6 / 5 = 6/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 6x^4 dx = 6/5x^5 + C |
Find the roots of the quadratic equation: 7x^2 + -6x + -10 = 0 | 1. Identify coefficients: a=7, b=-6, c=-10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 36 - -280 = 316.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (6 ± √316) / 14.
5. Calculate roots: x1 = 1.6983134881879411, x2 = -0.841170631045084.
Final... |
Evaluate tan(0°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(0°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.0000 (exact form depends on root simplification if applicable) |
In a right-angled triangle, the lengths of the two legs are a = 13 and b = 20. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 13^2 + 20^2 = c^2
2. Calculate the squares: 169 + 400 = c^2
3. Sum the squares: 569 = c^2
4. Take the square root of both sides: c = √569
Final A... |
Find the derivative of f(x) = (3x + 2)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 3x + 2.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 2. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 2)^2 * 3 = 9(3x +... |
Find the derivative of f(x) = (3x + 3)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 3x + 3.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 3. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 3)^2 * 3 = 9(3x +... |
Calculate the mean, population variance, and standard deviation for the dataset: [20, 16, 18, 10, 4] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (20 + 16 + 18 + 10 + 4) / 5 = 68 / 5 = 13.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [6.4, 2.4, 4.4, -3.6, -9.6]
Squared Differences: [40.96, 5.76, 19.36, 12.96, 92.16]... |
Find the roots of the quadratic equation: 9x^2 + 2x + -2 = 0 | 1. Identify coefficients: a=9, b=2, c=-2.
2. Calculate the discriminant: Δ = b^2 - 4ac = 4 - -72 = 76.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-2 ± √76) / 18.
5. Calculate roots: x1 = 0.37321099372674155, x2 = -0.5954332159489638.
Final An... |
Find the derivative of f(x) = (2x + 2)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 2x + 2.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 2x + 2. Its derivative is h'(x) = 2.
4. Multiply g'(u) by h'(x): f'(x) = 3(2x + 2)^2 * 2 = 6(2x +... |
Evaluate sin(60°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(60°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.8660 (exact form depends on root simplification if applicable) |
Find the roots of the quadratic equation: 3x^2 + 1x + 1 = 0 | 1. Identify coefficients: a=3, b=1, c=1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1^2 - 4(3)(1) = 1 - 12 = -11.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-1 ± √11i) / 6.
Final Answer: x = -0.16666666666666666 ± 0.5527707983925666i |
In a right-angled triangle, the lengths of the two legs are a = 9 and b = 10. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 9^2 + 10^2 = c^2
2. Calculate the squares: 81 + 100 = c^2
3. Sum the squares: 181 = c^2
4. Take the square root of both sides: c = √181
Final Ans... |
Add the fractions 4/8 and 1/12. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 8 and 12, which is 24.
2. Convert 4/8 by multiplying top and bottom by 3: 12/24.
3. Convert 1/12 by multiplying top and bottom by 2: 2/24.
4. Add the numerators: 12 + 2 = 14.
5. The resulting fraction is 14/24.
6. Simplify by div... |
In a right-angled triangle, the lengths of the two legs are a = 10 and b = 8. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 10^2 + 8^2 = c^2
2. Calculate the squares: 100 + 64 = c^2
3. Sum the squares: 164 = c^2
4. Take the square root of both sides: c = √164
Final Ans... |
Compute the indefinite integral of f(x) = 5x^1. | To find the indefinite integral ∫ 5x^1 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 1 + 1 = 2.
2. Divide the coefficient by the new exponent: 5 / 2 = 5/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 5x^1 dx = 5/2x^2 + C |
Compute the indefinite integral of f(x) = 1x^1. | To find the indefinite integral ∫ 1x^1 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 1 + 1 = 2.
2. Divide the coefficient by the new exponent: 1 / 2 = 1/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 1x^1 dx = 1/2x^2 + C |
Find the derivative of f(x) = (2x + 4)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 2x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 2x + 4. Its derivative is h'(x) = 2.
4. Multiply g'(u) by h'(x): f'(x) = 3(2x + 4)^2 * 2 = 6(2x +... |
Multiply the following 2x2 matrices:
A =
[-1, -2]
[-3, 5]
B =
[-3, -3]
[-2, 3] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-1*-3) + (-2*-2) = 7
C12 = (A11 * B12) + (A12 * B22) = (-1*-3) + (-2*3) = -3
C21 = (A21 * B11) + (A22 * B21) = (-3*-3) + (5*-2) = -1
C22 = (A21 * B12) + (A22 * B22) = (-3*-3) + (5*3) = 24
Re... |
Find the roots of the quadratic equation: 1x^2 + 7x + -10 = 0 | 1. Identify coefficients: a=1, b=7, c=-10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 49 - -40 = 89.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-7 ± √89) / 2.
5. Calculate roots: x1 = 1.2169905660283016, x2 = -8.2169905660283.
Final Answe... |
In a right-angled triangle, the lengths of the two legs are a = 15 and b = 13. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 15^2 + 13^2 = c^2
2. Calculate the squares: 225 + 169 = c^2
3. Sum the squares: 394 = c^2
4. Take the square root of both sides: c = √394
Final A... |
Multiply the following 2x2 matrices:
A =
[4, 2]
[-3, -2]
B =
[-5, -2]
[5, 1] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (4*-5) + (2*5) = -10
C12 = (A11 * B12) + (A12 * B22) = (4*-2) + (2*1) = -6
C21 = (A21 * B11) + (A22 * B21) = (-3*-5) + (-2*5) = 5
C22 = (A21 * B12) + (A22 * B22) = (-3*-2) + (-2*1) = 4
Result... |
Find the roots of the quadratic equation: 3x^2 + -8x + -1 = 0 | 1. Identify coefficients: a=3, b=-8, c=-1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 64 - -12 = 76.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (8 ± √76) / 6.
5. Calculate roots: x1 = 2.786299647846891, x2 = -0.11963298118022465.
Final Ans... |
Compute the indefinite integral of f(x) = 7x^5. | To find the indefinite integral ∫ 7x^5 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 5 + 1 = 6.
2. Divide the coefficient by the new exponent: 7 / 6 = 7/6.
3. Combine and add the constant of integration C.
Final Answer: ∫ 7x^5 dx = 7/6x^6 + C |
Add the fractions 12/11 and 12/7. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 11 and 7, which is 77.
2. Convert 12/11 by multiplying top and bottom by 7: 84/77.
3. Convert 12/7 by multiplying top and bottom by 11: 132/77.
4. Add the numerators: 84 + 132 = 216.
5. The resulting fraction is 216/77.
Final Ans... |
Multiply the following 2x2 matrices:
A =
[-1, 5]
[4, 5]
B =
[3, 4]
[1, 0] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-1*3) + (5*1) = 2
C12 = (A11 * B12) + (A12 * B22) = (-1*4) + (5*0) = -4
C21 = (A21 * B11) + (A22 * B21) = (4*3) + (5*1) = 17
C22 = (A21 * B12) + (A22 * B22) = (4*4) + (5*0) = 16
Result Matri... |
Evaluate the logarithm: log_4(64) | To evaluate log_4(64), we ask the question: '4 raised to what power equals 64?'
Let x be the unknown power: 4^x = 64
Since 4^3 = 64, it follows that x = 3.
Final Answer: 3 |
Multiply the following 2x2 matrices:
A =
[2, -5]
[1, 1]
B =
[-1, 3]
[-3, -5] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (2*-1) + (-5*-3) = 13
C12 = (A11 * B12) + (A12 * B22) = (2*3) + (-5*-5) = 31
C21 = (A21 * B11) + (A22 * B21) = (1*-1) + (1*-3) = -4
C22 = (A21 * B12) + (A22 * B22) = (1*3) + (1*-5) = -2
Resul... |
Evaluate sin(45°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(45°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.7071 (exact form depends on root simplification if applicable) |
In a right-angled triangle, the lengths of the two legs are a = 10 and b = 20. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 10^2 + 20^2 = c^2
2. Calculate the squares: 100 + 400 = c^2
3. Sum the squares: 500 = c^2
4. Take the square root of both sides: c = √500
Final A... |
Find the roots of the quadratic equation: 7x^2 + -2x + 7 = 0 | 1. Identify coefficients: a=7, b=-2, c=7.
2. Calculate the discriminant: Δ = b^2 - 4ac = -2^2 - 4(7)(7) = 4 - 196 = -192.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (2 ± √192i) / 14.
Final Answer: x = 0.14285714285714285 ± 0.989743318610787i |
Multiply the following 2x2 matrices:
A =
[-1, 2]
[-2, -5]
B =
[0, -1]
[-4, 0] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-1*0) + (2*-4) = -8
C12 = (A11 * B12) + (A12 * B22) = (-1*-1) + (2*0) = 1
C21 = (A21 * B11) + (A22 * B21) = (-2*0) + (-5*-4) = 20
C22 = (A21 * B12) + (A22 * B22) = (-2*-1) + (-5*0) = 2
Resul... |
Evaluate the logarithm: log_3(27) | To evaluate log_3(27), we ask the question: '3 raised to what power equals 27?'
Let x be the unknown power: 3^x = 27
Since 3^3 = 27, it follows that x = 3.
Final Answer: 3 |
Compute the quotient and remainder of 14152 divided by 46. Show the long division steps. | To divide 14152 by 46:
1. Determine how many times 46 goes into the leading digits of 14152.
2. Multiply the divisor by this estimate, subtract from the current dividend portion, and bring down the next digit.
3. Repeat until all digits are processed.
Result: 14152 / 46 = 307 with a remainder of 30.
Proof: (307 * 46) ... |
Compute the indefinite integral of f(x) = 7x^5. | To find the indefinite integral ∫ 7x^5 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 5 + 1 = 6.
2. Divide the coefficient by the new exponent: 7 / 6 = 7/6.
3. Combine and add the constant of integration C.
Final Answer: ∫ 7x^5 dx = 7/6x^6 + C |
Compute the indefinite integral of f(x) = 2x^4. | To find the indefinite integral ∫ 2x^4 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 4 + 1 = 5.
2. Divide the coefficient by the new exponent: 2 / 5 = 2/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 2x^4 dx = 2/5x^5 + C |
Calculate the mean, population variance, and standard deviation for the dataset: [10, 13, 2, 6, 4] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (10 + 13 + 2 + 6 + 4) / 5 = 35 / 5 = 7.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [3.0, 6.0, -5.0, -1.0, -3.0]
Squared Differences: [9.0, 36.0, 25.0, 1.0, 9.0]
Sum o... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.