prompt stringlengths 32 115 | completion stringlengths 167 566 |
|---|---|
Compute the indefinite integral of f(x) = 8x^4. | To find the indefinite integral ∫ 8x^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: 8 / 5 = 8/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 8x^4 dx = 8/5x^5 + C |
Calculate the mean, population variance, and standard deviation for the dataset: [4, 16, 4, 14, 1] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (4 + 16 + 4 + 14 + 1) / 5 = 39 / 5 = 7.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-3.8, 8.2, -3.8, 6.2, -6.8]
Squared Differences: [14.44, 67.24, 14.44, 38.44, 46.24]
... |
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) |
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 |
Calculate the mean, population variance, and standard deviation for the dataset: [9, 16, 17, 19, 13] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (9 + 16 + 17 + 19 + 13) / 5 = 74 / 5 = 14.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-5.8, 1.2, 2.2, 4.2, -1.8]
Squared Differences: [33.64, 1.44, 4.84, 17.64, 3.24]
... |
Find the roots of the quadratic equation: 10x^2 + -3x + -5 = 0 | 1. Identify coefficients: a=10, b=-3, c=-5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 9 - -200 = 209.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (3 ± √209) / 20.
5. Calculate roots: x1 = 0.8728416147400481, x2 = -0.572841614740048.
Final ... |
Compute the quotient and remainder of 8774 divided by 15. Show the long division steps. | To divide 8774 by 15:
1. Determine how many times 15 goes into the leading digits of 8774.
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: 8774 / 15 = 584 with a remainder of 14.
Proof: (584 * 15) + 1... |
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 |
Calculate the mean, population variance, and standard deviation for the dataset: [5, 17, 3, 15, 6] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (5 + 17 + 3 + 15 + 6) / 5 = 46 / 5 = 9.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-4.2, 7.8, -6.2, 5.8, -3.2]
Squared Differences: [17.64, 60.84, 38.44, 33.64, 10.24]
... |
Find the roots of the quadratic equation: 9x^2 + -6x + -10 = 0 | 1. Identify coefficients: a=9, b=-6, c=-10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 36 - -360 = 396.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (6 ± √396) / 18.
5. Calculate roots: x1 = 1.4388749301184667, x2 = -0.7722082634517999.
Fina... |
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 |
Find the derivative of f(x) = (3x + 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 = 3x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 5. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 5)^2 * 3 = 9(3x +... |
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 +... |
Calculate the mean, population variance, and standard deviation for the dataset: [8, 15, 20, 11, 7] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (8 + 15 + 20 + 11 + 7) / 5 = 61 / 5 = 12.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-4.2, 2.8, 7.8, -1.2, -5.2]
Squared Differences: [17.64, 7.84, 60.84, 1.44, 27.04]
... |
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 indefinite integral of f(x) = 6x^1. | To find the indefinite integral ∫ 6x^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: 6 / 2 = 3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 6x^1 dx = 3x^2 + C |
Calculate the mean, population variance, and standard deviation for the dataset: [4, 12, 16, 7, 7] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (4 + 12 + 16 + 7 + 7) / 5 = 46 / 5 = 9.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-5.2, 2.8, 6.8, -2.2, -2.2]
Squared Differences: [27.04, 7.84, 46.24, 4.84, 4.84]
... |
Multiply the following 2x2 matrices:
A =
[3, -4]
[2, 3]
B =
[-4, 0]
[5, 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) = (3*-4) + (-4*5) = -32
C12 = (A11 * B12) + (A12 * B22) = (3*0) + (-4*5) = -20
C21 = (A21 * B11) + (A22 * B21) = (2*-4) + (3*5) = 7
C22 = (A21 * B12) + (A22 * B22) = (2*0) + (3*5) = 15
Result M... |
Calculate the mean, population variance, and standard deviation for the dataset: [5, 12, 6, 10, 2] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (5 + 12 + 6 + 10 + 2) / 5 = 35 / 5 = 7.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-2.0, 5.0, -1.0, 3.0, -5.0]
Squared Differences: [4.0, 25.0, 1.0, 9.0, 25.0]
Sum o... |
Compute the indefinite integral of f(x) = 1x^2. | To find the indefinite integral ∫ 1x^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: 1 / 3 = 1/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 1x^2 dx = 1/3x^3 + C |
Calculate the mean, population variance, and standard deviation for the dataset: [1, 20, 20, 14, 5] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (1 + 20 + 20 + 14 + 5) / 5 = 60 / 5 = 12.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-11.0, 8.0, 8.0, 2.0, -7.0]
Squared Differences: [121.0, 64.0, 64.0, 4.0, 49.0]
... |
Calculate the mean, population variance, and standard deviation for the dataset: [1, 2, 1, 13, 17] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (1 + 2 + 1 + 13 + 17) / 5 = 34 / 5 = 6.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-5.8, -4.8, -5.8, 6.2, 10.2]
Squared Differences: [33.64, 23.04, 33.64, 38.44, 104.04... |
Compute the quotient and remainder of 16747 divided by 91. Show the long division steps. | To divide 16747 by 91:
1. Determine how many times 91 goes into the leading digits of 16747.
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: 16747 / 91 = 184 with a remainder of 3.
Proof: (184 * 91) +... |
Find the roots of the quadratic equation: 1x^2 + -5x + -2 = 0 | 1. Identify coefficients: a=1, b=-5, c=-2.
2. Calculate the discriminant: Δ = b^2 - 4ac = 25 - -8 = 33.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (5 ± √33) / 2.
5. Calculate roots: x1 = 5.372281323269014, x2 = -0.3722813232690143.
Final Answe... |
Find the roots of the quadratic equation: 5x^2 + -4x + -9 = 0 | 1. Identify coefficients: a=5, b=-4, c=-9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 16 - -180 = 196.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (4 ± √196) / 10.
5. Calculate roots: x1 = 1.8, x2 = -1.0.
Final Answer: x = 1.8, -1.0 |
Compute the quotient and remainder of 37986 divided by 87. Show the long division steps. | To divide 37986 by 87:
1. Determine how many times 87 goes into the leading digits of 37986.
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: 37986 / 87 = 436 with a remainder of 54.
Proof: (436 * 87) ... |
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 +... |
In a right-angled triangle, the lengths of the two legs are a = 9 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: 9^2 + 18^2 = c^2
2. Calculate the squares: 81 + 324 = c^2
3. Sum the squares: 405 = c^2
4. Take the square root of both sides: c = √405
Final Ans... |
Compute the quotient and remainder of 91790 divided by 23. Show the long division steps. | To divide 91790 by 23:
1. Determine how many times 23 goes into the leading digits of 91790.
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: 91790 / 23 = 3990 with a remainder of 20.
Proof: (3990 * 23... |
In a right-angled triangle, the lengths of the two legs are a = 12 and b = 11. 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 + 11^2 = c^2
2. Calculate the squares: 144 + 121 = c^2
3. Sum the squares: 265 = c^2
4. Take the square root of both sides: c = √265
Final A... |
Calculate the mean, population variance, and standard deviation for the dataset: [14, 7, 2, 1, 18] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (14 + 7 + 2 + 1 + 18) / 5 = 42 / 5 = 8.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [5.6, -1.4, -6.4, -7.4, 9.6]
Squared Differences: [31.36, 1.96, 40.96, 54.76, 92.16]
... |
In a right-angled triangle, the lengths of the two legs are a = 5 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: 5^2 + 19^2 = c^2
2. Calculate the squares: 25 + 361 = c^2
3. Sum the squares: 386 = c^2
4. Take the square root of both sides: c = √386
Final Ans... |
Add the fractions 13/4 and 5/6. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 4 and 6, which is 12.
2. Convert 13/4 by multiplying top and bottom by 3: 39/12.
3. Convert 5/6 by multiplying top and bottom by 2: 10/12.
4. Add the numerators: 39 + 10 = 49.
5. The resulting fraction is 49/12.
Final Answer: 49/... |
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 ... |
Multiply the following 2x2 matrices:
A =
[-2, -3]
[-3, -5]
B =
[5, -4]
[3, 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*5) + (-3*3) = -19
C12 = (A11 * B12) + (A12 * B22) = (-2*-4) + (-3*3) = -1
C21 = (A21 * B11) + (A22 * B21) = (-3*5) + (-5*3) = -30
C22 = (A21 * B12) + (A22 * B22) = (-3*-4) + (-5*3) = -3
R... |
Evaluate the logarithm: log_2(32) | To evaluate log_2(32), we ask the question: '2 raised to what power equals 32?'
Let x be the unknown power: 2^x = 32
Since 2^5 = 32, it follows that x = 5.
Final Answer: 5 |
Multiply the following 2x2 matrices:
A =
[-5, -4]
[-4, -5]
B =
[-3, 3]
[2, -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) = (-5*-3) + (-4*2) = 7
C12 = (A11 * B12) + (A12 * B22) = (-5*3) + (-4*-2) = -7
C21 = (A21 * B11) + (A22 * B21) = (-4*-3) + (-5*2) = 2
C22 = (A21 * B12) + (A22 * B22) = (-4*3) + (-5*-2) = -2
Res... |
Evaluate the logarithm: log_3(9) | To evaluate log_3(9), we ask the question: '3 raised to what power equals 9?'
Let x be the unknown power: 3^x = 9
Since 3^2 = 9, it follows that x = 2.
Final Answer: 2 |
Multiply the following 2x2 matrices:
A =
[-2, -3]
[-2, -5]
B =
[-3, 4]
[4, -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) = (-2*-3) + (-3*4) = -6
C12 = (A11 * B12) + (A12 * B22) = (-2*4) + (-3*-1) = -5
C21 = (A21 * B11) + (A22 * B21) = (-2*-3) + (-5*4) = -14
C22 = (A21 * B12) + (A22 * B22) = (-2*4) + (-5*-1) = -3
... |
Add the fractions 3/12 and 11/13. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 12 and 13, which is 156.
2. Convert 3/12 by multiplying top and bottom by 13: 39/156.
3. Convert 11/13 by multiplying top and bottom by 12: 132/156.
4. Add the numerators: 39 + 132 = 171.
5. The resulting fraction is 171/156.
6. ... |
Find the roots of the quadratic equation: 4x^2 + -9x + -5 = 0 | 1. Identify coefficients: a=4, b=-9, c=-5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 81 - -80 = 161.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (9 ± √161) / 8.
5. Calculate roots: x1 = 2.7110721925561903, x2 = -0.46107219255619003.
Final ... |
Compute the quotient and remainder of 11265 divided by 65. Show the long division steps. | To divide 11265 by 65:
1. Determine how many times 65 goes into the leading digits of 11265.
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: 11265 / 65 = 173 with a remainder of 20.
Proof: (173 * 65) ... |
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 +... |
Compute the quotient and remainder of 56064 divided by 85. Show the long division steps. | To divide 56064 by 85:
1. Determine how many times 85 goes into the leading digits of 56064.
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: 56064 / 85 = 659 with a remainder of 49.
Proof: (659 * 85) ... |
Evaluate tan(30°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(30°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.5774 (exact form depends on root simplification if applicable) |
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) |
Evaluate the logarithm: log_4(16) | To evaluate log_4(16), we ask the question: '4 raised to what power equals 16?'
Let x be the unknown power: 4^x = 16
Since 4^2 = 16, it follows that x = 2.
Final Answer: 2 |
Find the roots of the quadratic equation: 8x^2 + -1x + 4 = 0 | 1. Identify coefficients: a=8, b=-1, c=4.
2. Calculate the discriminant: Δ = b^2 - 4ac = -1^2 - 4(8)(4) = 1 - 128 = -127.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (1 ± √127i) / 16.
Final Answer: x = 0.0625 ± 0.7043392293490403i |
Evaluate the logarithm: log_2(32) | To evaluate log_2(32), we ask the question: '2 raised to what power equals 32?'
Let x be the unknown power: 2^x = 32
Since 2^5 = 32, it follows that x = 5.
Final Answer: 5 |
Compute the quotient and remainder of 98121 divided by 75. Show the long division steps. | To divide 98121 by 75:
1. Determine how many times 75 goes into the leading digits of 98121.
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: 98121 / 75 = 1308 with a remainder of 21.
Proof: (1308 * 75... |
Add the fractions 9/12 and 1/11. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 12 and 11, which is 132.
2. Convert 9/12 by multiplying top and bottom by 11: 99/132.
3. Convert 1/11 by multiplying top and bottom by 12: 12/132.
4. Add the numerators: 99 + 12 = 111.
5. The resulting fraction is 111/132.
6. Sim... |
Find the roots of the quadratic equation: 4x^2 + 8x + 0 = 0 | 1. Identify coefficients: a=4, b=8, c=0.
2. Calculate the discriminant: Δ = b^2 - 4ac = 64 - 0 = 64.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-8 ± √64) / 8.
5. Calculate roots: x1 = 0.0, x2 = -2.0.
Final Answer: x = 0.0, -2.0 |
Add the fractions 5/14 and 1/4. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 14 and 4, which is 28.
2. Convert 5/14 by multiplying top and bottom by 2: 10/28.
3. Convert 1/4 by multiplying top and bottom by 7: 7/28.
4. Add the numerators: 10 + 7 = 17.
5. The resulting fraction is 17/28.
Final Answer: 17/2... |
Multiply the following 2x2 matrices:
A =
[5, -3]
[-1, 4]
B =
[3, 1]
[-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) = (5*3) + (-3*-4) = 27
C12 = (A11 * B12) + (A12 * B22) = (5*1) + (-3*5) = -10
C21 = (A21 * B11) + (A22 * B21) = (-1*3) + (4*-4) = -19
C22 = (A21 * B12) + (A22 * B22) = (-1*1) + (4*5) = 19
Resul... |
Find the roots of the quadratic equation: 6x^2 + -7x + -10 = 0 | 1. Identify coefficients: a=6, b=-7, c=-10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 49 - -240 = 289.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (7 ± √289) / 12.
5. Calculate roots: x1 = 2.0, x2 = -0.8333333333333334.
Final Answer: x = 2... |
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 +... |
Evaluate the logarithm: log_2(32) | To evaluate log_2(32), we ask the question: '2 raised to what power equals 32?'
Let x be the unknown power: 2^x = 32
Since 2^5 = 32, it follows that x = 5.
Final Answer: 5 |
Compute the quotient and remainder of 76836 divided by 31. Show the long division steps. | To divide 76836 by 31:
1. Determine how many times 31 goes into the leading digits of 76836.
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: 76836 / 31 = 2478 with a remainder of 18.
Proof: (2478 * 31... |
Evaluate the logarithm: log_5(3125) | To evaluate log_5(3125), we ask the question: '5 raised to what power equals 3125?'
Let x be the unknown power: 5^x = 3125
Since 5^5 = 3125, it follows that x = 5.
Final Answer: 5 |
Compute the indefinite integral of f(x) = 10x^4. | To find the indefinite integral ∫ 10x^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: 10 / 5 = 2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 10x^4 dx = 2x^5 + C |
In a right-angled triangle, the lengths of the two legs are a = 15 and b = 12. 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 + 12^2 = c^2
2. Calculate the squares: 225 + 144 = c^2
3. Sum the squares: 369 = c^2
4. Take the square root of both sides: c = √369
Final A... |
Find the derivative of f(x) = (3x + 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 = 3x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 4. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 4)^2 * 3 = 9(3x +... |
Compute the quotient and remainder of 32455 divided by 98. Show the long division steps. | To divide 32455 by 98:
1. Determine how many times 98 goes into the leading digits of 32455.
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: 32455 / 98 = 331 with a remainder of 17.
Proof: (331 * 98) ... |
Multiply the following 2x2 matrices:
A =
[-4, 0]
[1, -4]
B =
[-3, 5]
[0, 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) = (-4*-3) + (0*0) = 12
C12 = (A11 * B12) + (A12 * B22) = (-4*5) + (0*0) = -20
C21 = (A21 * B11) + (A22 * B21) = (1*-3) + (-4*0) = -3
C22 = (A21 * B12) + (A22 * B22) = (1*5) + (-4*0) = 5
Result ... |
Calculate the mean, population variance, and standard deviation for the dataset: [15, 9, 12, 13, 2] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (15 + 9 + 12 + 13 + 2) / 5 = 51 / 5 = 10.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [4.8, -1.2, 1.8, 2.8, -8.2]
Squared Differences: [23.04, 1.44, 3.24, 7.84, 67.24]
... |
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 |
Calculate the mean, population variance, and standard deviation for the dataset: [9, 17, 1, 3, 6] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (9 + 17 + 1 + 3 + 6) / 5 = 36 / 5 = 7.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [1.8, 9.8, -6.2, -4.2, -1.2]
Squared Differences: [3.24, 96.04, 38.44, 17.64, 1.44]
... |
Evaluate sin(0°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(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) |
Evaluate the logarithm: log_3(81) | To evaluate log_3(81), we ask the question: '3 raised to what power equals 81?'
Let x be the unknown power: 3^x = 81
Since 3^4 = 81, it follows that x = 4.
Final Answer: 4 |
Add the fractions 5/12 and 12/12. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 12 and 12, which is 12.
2. Convert 5/12 by multiplying top and bottom by 1: 5/12.
3. Convert 12/12 by multiplying top and bottom by 1: 12/12.
4. Add the numerators: 5 + 12 = 17.
5. The resulting fraction is 17/12.
Final Answer: 1... |
Add the fractions 15/6 and 3/5. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 6 and 5, which is 30.
2. Convert 15/6 by multiplying top and bottom by 5: 75/30.
3. Convert 3/5 by multiplying top and bottom by 6: 18/30.
4. Add the numerators: 75 + 18 = 93.
5. The resulting fraction is 93/30.
6. Simplify by di... |
Compute the quotient and remainder of 6292 divided by 85. Show the long division steps. | To divide 6292 by 85:
1. Determine how many times 85 goes into the leading digits of 6292.
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: 6292 / 85 = 74 with a remainder of 2.
Proof: (74 * 85) + 2 = ... |
Add the fractions 2/11 and 15/10. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 11 and 10, which is 110.
2. Convert 2/11 by multiplying top and bottom by 10: 20/110.
3. Convert 15/10 by multiplying top and bottom by 11: 165/110.
4. Add the numerators: 20 + 165 = 185.
5. The resulting fraction is 185/110.
6. ... |
Evaluate tan(30°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(30°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.5774 (exact form depends on root simplification if applicable) |
Compute the indefinite integral of f(x) = 6x^1. | To find the indefinite integral ∫ 6x^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: 6 / 2 = 3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 6x^1 dx = 3x^2 + C |
Evaluate tan(90°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(90°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 16331239353195370.0000 (exact form depends on root simplification if applicable) |
Evaluate the logarithm: log_4(16) | To evaluate log_4(16), we ask the question: '4 raised to what power equals 16?'
Let x be the unknown power: 4^x = 16
Since 4^2 = 16, it follows that x = 2.
Final Answer: 2 |
Calculate the mean, population variance, and standard deviation for the dataset: [6, 9, 13, 2, 15] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (6 + 9 + 13 + 2 + 15) / 5 = 45 / 5 = 9.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-3.0, 0.0, 4.0, -7.0, 6.0]
Squared Differences: [9.0, 0.0, 16.0, 49.0, 36.0]
Sum o... |
Evaluate tan(30°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(30°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.5774 (exact form depends on root simplification if applicable) |
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 ... |
Add the fractions 9/3 and 13/13. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 3 and 13, which is 39.
2. Convert 9/3 by multiplying top and bottom by 13: 117/39.
3. Convert 13/13 by multiplying top and bottom by 3: 39/39.
4. Add the numerators: 117 + 39 = 156.
5. The resulting fraction is 156/39.
6. Simplif... |
Add the fractions 1/12 and 3/10. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 12 and 10, which is 60.
2. Convert 1/12 by multiplying top and bottom by 5: 5/60.
3. Convert 3/10 by multiplying top and bottom by 6: 18/60.
4. Add the numerators: 5 + 18 = 23.
5. The resulting fraction is 23/60.
Final Answer: 23... |
Calculate the mean, population variance, and standard deviation for the dataset: [15, 19, 8, 7, 8] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (15 + 19 + 8 + 7 + 8) / 5 = 57 / 5 = 11.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [3.6, 7.6, -3.4, -4.4, -3.4]
Squared Differences: [12.96, 57.76, 11.56, 19.36, 11.56]... |
Add the fractions 14/12 and 2/8. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 12 and 8, which is 24.
2. Convert 14/12 by multiplying top and bottom by 2: 28/24.
3. Convert 2/8 by multiplying top and bottom by 3: 6/24.
4. Add the numerators: 28 + 6 = 34.
5. The resulting fraction is 34/24.
6. Simplify by di... |
Add the fractions 8/4 and 6/4. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 4 and 4, which is 4.
2. Convert 8/4 by multiplying top and bottom by 1: 8/4.
3. Convert 6/4 by multiplying top and bottom by 1: 6/4.
4. Add the numerators: 8 + 6 = 14.
5. The resulting fraction is 14/4.
6. Simplify by dividing nu... |
Add the fractions 12/4 and 7/2. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 4 and 2, which is 4.
2. Convert 12/4 by multiplying top and bottom by 1: 12/4.
3. Convert 7/2 by multiplying top and bottom by 2: 14/4.
4. Add the numerators: 12 + 14 = 26.
5. The resulting fraction is 26/4.
6. Simplify by dividi... |
Multiply the following 2x2 matrices:
A =
[-5, 2]
[1, 3]
B =
[-4, 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) = (-5*-4) + (2*2) = 24
C12 = (A11 * B12) + (A12 * B22) = (-5*3) + (2*-3) = -21
C21 = (A21 * B11) + (A22 * B21) = (1*-4) + (3*2) = 2
C22 = (A21 * B12) + (A22 * B22) = (1*3) + (3*-3) = -6
Result ... |
Multiply the following 2x2 matrices:
A =
[-4, 3]
[1, 4]
B =
[1, 4]
[-1, -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) = (-4*1) + (3*-1) = -7
C12 = (A11 * B12) + (A12 * B22) = (-4*4) + (3*-5) = -31
C21 = (A21 * B11) + (A22 * B21) = (1*1) + (4*-1) = -3
C22 = (A21 * B12) + (A22 * B22) = (1*4) + (4*-5) = -16
Resul... |
Calculate the mean, population variance, and standard deviation for the dataset: [2, 2, 19, 20, 13] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (2 + 2 + 19 + 20 + 13) / 5 = 56 / 5 = 11.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-9.2, -9.2, 7.8, 8.8, 1.8]
Squared Differences: [84.64, 84.64, 60.84, 77.44, 3.24]
... |
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 |
Multiply the following 2x2 matrices:
A =
[3, -2]
[4, 2]
B =
[0, -5]
[5, -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*0) + (-2*5) = -10
C12 = (A11 * B12) + (A12 * B22) = (3*-5) + (-2*-3) = -9
C21 = (A21 * B11) + (A22 * B21) = (4*0) + (2*5) = 10
C22 = (A21 * B12) + (A22 * B22) = (4*-5) + (2*-3) = -26
Resul... |
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) |
Compute the indefinite integral of f(x) = 2x^2. | To find the indefinite integral ∫ 2x^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: 2 / 3 = 2/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 2x^2 dx = 2/3x^3 + C |
In a right-angled triangle, the lengths of the two legs are a = 17 and b = 17. 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: 17^2 + 17^2 = c^2
2. Calculate the squares: 289 + 289 = c^2
3. Sum the squares: 578 = c^2
4. Take the square root of both sides: c = √578
Final A... |
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) |
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 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 ... |
Compute the quotient and remainder of 46525 divided by 72. Show the long division steps. | To divide 46525 by 72:
1. Determine how many times 72 goes into the leading digits of 46525.
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: 46525 / 72 = 646 with a remainder of 13.
Proof: (646 * 72) ... |
In a right-angled triangle, the lengths of the two legs are a = 8 and b = 16. 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 + 16^2 = c^2
2. Calculate the squares: 64 + 256 = c^2
3. Sum the squares: 320 = c^2
4. Take the square root of both sides: c = √320
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) |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.