prompt stringlengths 32 115 | completion stringlengths 167 566 |
|---|---|
In a right-angled triangle, the lengths of the two legs are a = 5 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: 5^2 + 20^2 = c^2
2. Calculate the squares: 25 + 400 = c^2
3. Sum the squares: 425 = c^2
4. Take the square root of both sides: c = √425
Final Ans... |
Compute the quotient and remainder of 2818 divided by 27. Show the long division steps. | To divide 2818 by 27:
1. Determine how many times 27 goes into the leading digits of 2818.
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: 2818 / 27 = 104 with a remainder of 10.
Proof: (104 * 27) + 1... |
In a right-angled triangle, the lengths of the two legs are a = 5 and b = 15. 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 + 15^2 = c^2
2. Calculate the squares: 25 + 225 = c^2
3. Sum the squares: 250 = c^2
4. Take the square root of both sides: c = √250
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 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 quotient and remainder of 65510 divided by 10. Show the long division steps. | To divide 65510 by 10:
1. Determine how many times 10 goes into the leading digits of 65510.
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: 65510 / 10 = 6551 with a remainder of 0.
Proof: (6551 * 10)... |
Compute the indefinite integral of f(x) = 8x^1. | To find the indefinite integral ∫ 8x^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: 8 / 2 = 4.
3. Combine and add the constant of integration C.
Final Answer: ∫ 8x^1 dx = 4x^2 + C |
Multiply the following 2x2 matrices:
A =
[1, -3]
[5, 4]
B =
[-4, -2]
[-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) = (1*-4) + (-3*-3) = 5
C12 = (A11 * B12) + (A12 * B22) = (1*-2) + (-3*-5) = 13
C21 = (A21 * B11) + (A22 * B21) = (5*-4) + (4*-3) = -32
C22 = (A21 * B12) + (A22 * B22) = (5*-2) + (4*-5) = -30
Re... |
In a right-angled triangle, the lengths of the two legs are a = 16 and b = 3. 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 + 3^2 = c^2
2. Calculate the squares: 256 + 9 = c^2
3. Sum the squares: 265 = c^2
4. Take the square root of both sides: c = √265
Final Answ... |
Find the roots of the quadratic equation: 9x^2 + 8x + 9 = 0 | 1. Identify coefficients: a=9, b=8, c=9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 8^2 - 4(9)(9) = 64 - 324 = -260.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-8 ± √260i) / 18.
Final Answer: x = -0.4444444444444444 ± 0.8958064164776166i |
Calculate the mean, population variance, and standard deviation for the dataset: [9, 20, 19, 20, 12] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (9 + 20 + 19 + 20 + 12) / 5 = 80 / 5 = 16.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-7.0, 4.0, 3.0, 4.0, -4.0]
Squared Differences: [49.0, 16.0, 9.0, 16.0, 16.0]
S... |
Multiply the following 2x2 matrices:
A =
[5, -5]
[0, 1]
B =
[-5, 5]
[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) = (5*-5) + (-5*3) = -40
C12 = (A11 * B12) + (A12 * B22) = (5*5) + (-5*5) = 0
C21 = (A21 * B11) + (A22 * B21) = (0*-5) + (1*3) = 3
C22 = (A21 * B12) + (A22 * B22) = (0*5) + (1*5) = 5
Result Matr... |
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 |
Evaluate cos(60°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(60°) 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) |
Compute the indefinite integral of f(x) = 9x^2. | To find the indefinite integral ∫ 9x^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: 9 / 3 = 3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 9x^2 dx = 3x^3 + C |
In a right-angled triangle, the lengths of the two legs are a = 18 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: 18^2 + 16^2 = c^2
2. Calculate the squares: 324 + 256 = c^2
3. Sum the squares: 580 = c^2
4. Take the square root of both sides: c = √580
Final A... |
Add the fractions 6/12 and 10/13. 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 6/12 by multiplying top and bottom by 13: 78/156.
3. Convert 10/13 by multiplying top and bottom by 12: 120/156.
4. Add the numerators: 78 + 120 = 198.
5. The resulting fraction is 198/156.
6. ... |
Multiply the following 2x2 matrices:
A =
[-2, 1]
[-5, -5]
B =
[5, -5]
[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) = (-2*5) + (1*1) = -9
C12 = (A11 * B12) + (A12 * B22) = (-2*-5) + (1*5) = 15
C21 = (A21 * B11) + (A22 * B21) = (-5*5) + (-5*1) = -30
C22 = (A21 * B12) + (A22 * B22) = (-5*-5) + (-5*5) = 0
Resul... |
Compute the quotient and remainder of 3727 divided by 89. Show the long division steps. | To divide 3727 by 89:
1. Determine how many times 89 goes into the leading digits of 3727.
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: 3727 / 89 = 41 with a remainder of 78.
Proof: (41 * 89) + 78 ... |
Compute the indefinite integral of f(x) = 8x^2. | To find the indefinite integral ∫ 8x^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: 8 / 3 = 8/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 8x^2 dx = 8/3x^3 + C |
Evaluate cos(0°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(0°) 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) |
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 ... |
Add the fractions 9/10 and 6/15. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 10 and 15, which is 30.
2. Convert 9/10 by multiplying top and bottom by 3: 27/30.
3. Convert 6/15 by multiplying top and bottom by 2: 12/30.
4. Add the numerators: 27 + 12 = 39.
5. The resulting fraction is 39/30.
6. Simplify by... |
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 +... |
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 = 20 and b = 3. 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: 20^2 + 3^2 = c^2
2. Calculate the squares: 400 + 9 = c^2
3. Sum the squares: 409 = c^2
4. Take the square root of both sides: c = √409
Final Answ... |
Find the roots of the quadratic equation: 10x^2 + 4x + -3 = 0 | 1. Identify coefficients: a=10, b=4, c=-3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 16 - -120 = 136.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-4 ± √136) / 20.
5. Calculate roots: x1 = 0.3830951894845301, x2 = -0.78309518948453.
Final ... |
Evaluate the logarithm: log_2(4) | To evaluate log_2(4), we ask the question: '2 raised to what power equals 4?'
Let x be the unknown power: 2^x = 4
Since 2^2 = 4, it follows that x = 2.
Final Answer: 2 |
Compute the quotient and remainder of 87307 divided by 3. Show the long division steps. | To divide 87307 by 3:
1. Determine how many times 3 goes into the leading digits of 87307.
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: 87307 / 3 = 29102 with a remainder of 1.
Proof: (29102 * 3) +... |
Multiply the following 2x2 matrices:
A =
[4, -2]
[-3, 3]
B =
[-2, -2]
[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) = (4*-2) + (-2*3) = -14
C12 = (A11 * B12) + (A12 * B22) = (4*-2) + (-2*-5) = 2
C21 = (A21 * B11) + (A22 * B21) = (-3*-2) + (3*3) = 15
C22 = (A21 * B12) + (A22 * B22) = (-3*-2) + (3*-5) = -9
Res... |
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) |
Calculate the mean, population variance, and standard deviation for the dataset: [2, 4, 4, 7, 3] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (2 + 4 + 4 + 7 + 3) / 5 = 20 / 5 = 4.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-2.0, 0.0, 0.0, 3.0, -1.0]
Squared Differences: [4.0, 0.0, 0.0, 9.0, 1.0]
Sum of Squ... |
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 |
Calculate the mean, population variance, and standard deviation for the dataset: [1, 11, 19, 13, 17] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (1 + 11 + 19 + 13 + 17) / 5 = 61 / 5 = 12.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-11.2, -1.2, 6.8, 0.8, 4.8]
Squared Differences: [125.44, 1.44, 46.24, 0.64, 23.04... |
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 |
Evaluate the logarithm: log_2(4) | To evaluate log_2(4), we ask the question: '2 raised to what power equals 4?'
Let x be the unknown power: 2^x = 4
Since 2^2 = 4, it follows that x = 2.
Final Answer: 2 |
Multiply the following 2x2 matrices:
A =
[-2, 5]
[0, 0]
B =
[0, 1]
[-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) = (-2*0) + (5*-5) = -25
C12 = (A11 * B12) + (A12 * B22) = (-2*1) + (5*4) = 18
C21 = (A21 * B11) + (A22 * B21) = (0*0) + (0*-5) = 0
C22 = (A21 * B12) + (A22 * B22) = (0*1) + (0*4) = 0
Result Mat... |
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 +... |
Compute the quotient and remainder of 84184 divided by 83. Show the long division steps. | To divide 84184 by 83:
1. Determine how many times 83 goes into the leading digits of 84184.
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: 84184 / 83 = 1014 with a remainder of 22.
Proof: (1014 * 83... |
Find the derivative of f(x) = (2x + 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 = 2x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 2x + 5. Its derivative is h'(x) = 2.
4. Multiply g'(u) by h'(x): f'(x) = 3(2x + 5)^2 * 2 = 6(2x +... |
Multiply the following 2x2 matrices:
A =
[-5, -3]
[-4, -2]
B =
[-1, -1]
[-1, -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*-1) + (-3*-1) = 8
C12 = (A11 * B12) + (A12 * B22) = (-5*-1) + (-3*-2) = 11
C21 = (A21 * B11) + (A22 * B21) = (-4*-1) + (-2*-1) = 6
C22 = (A21 * B12) + (A22 * B22) = (-4*-1) + (-2*-2) = 8
... |
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 ... |
In a right-angled triangle, the lengths of the two legs are a = 20 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: 20^2 + 16^2 = c^2
2. Calculate the squares: 400 + 256 = c^2
3. Sum the squares: 656 = c^2
4. Take the square root of both sides: c = √656
Final A... |
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 |
Evaluate the logarithm: log_2(4) | To evaluate log_2(4), we ask the question: '2 raised to what power equals 4?'
Let x be the unknown power: 2^x = 4
Since 2^2 = 4, it follows that x = 2.
Final Answer: 2 |
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 ... |
Multiply the following 2x2 matrices:
A =
[4, 0]
[-4, 4]
B =
[1, 1]
[2, -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) = (4*1) + (0*2) = 4
C12 = (A11 * B12) + (A12 * B22) = (4*1) + (0*-4) = 4
C21 = (A21 * B11) + (A22 * B21) = (-4*1) + (4*2) = 4
C22 = (A21 * B12) + (A22 * B22) = (-4*1) + (4*-4) = -20
Result Matr... |
Calculate the mean, population variance, and standard deviation for the dataset: [9, 7, 15, 9, 6] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (9 + 7 + 15 + 9 + 6) / 5 = 46 / 5 = 9.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-0.2, -2.2, 5.8, -0.2, -3.2]
Squared Differences: [0.04, 4.84, 33.64, 0.04, 10.24]
... |
Compute the indefinite integral of f(x) = 5x^5. | To find the indefinite integral ∫ 5x^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: 5 / 6 = 5/6.
3. Combine and add the constant of integration C.
Final Answer: ∫ 5x^5 dx = 5/6x^6 + C |
Compute the indefinite integral of f(x) = 5x^5. | To find the indefinite integral ∫ 5x^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: 5 / 6 = 5/6.
3. Combine and add the constant of integration C.
Final Answer: ∫ 5x^5 dx = 5/6x^6 + C |
Evaluate the logarithm: log_5(25) | To evaluate log_5(25), we ask the question: '5 raised to what power equals 25?'
Let x be the unknown power: 5^x = 25
Since 5^2 = 25, it follows that x = 2.
Final Answer: 2 |
Evaluate cos(60°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(60°) 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) |
Compute the quotient and remainder of 19488 divided by 4. Show the long division steps. | To divide 19488 by 4:
1. Determine how many times 4 goes into the leading digits of 19488.
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: 19488 / 4 = 4872 with a remainder of 0.
Proof: (4872 * 4) + 0... |
Find the derivative of f(x) = (2x + 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 = 2x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 2x + 5. Its derivative is h'(x) = 2.
4. Multiply g'(u) by h'(x): f'(x) = 3(2x + 5)^2 * 2 = 6(2x +... |
Calculate the mean, population variance, and standard deviation for the dataset: [10, 6, 12, 9, 18] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (10 + 6 + 12 + 9 + 18) / 5 = 55 / 5 = 11.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-1.0, -5.0, 1.0, -2.0, 7.0]
Squared Differences: [1.0, 25.0, 1.0, 4.0, 49.0]
Sum... |
Multiply the following 2x2 matrices:
A =
[-1, 3]
[-3, 2]
B =
[-5, 0]
[0, 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*-5) + (3*0) = 5
C12 = (A11 * B12) + (A12 * B22) = (-1*0) + (3*4) = 12
C21 = (A21 * B11) + (A22 * B21) = (-3*-5) + (2*0) = 15
C22 = (A21 * B12) + (A22 * B22) = (-3*0) + (2*4) = 8
Result Ma... |
Multiply the following 2x2 matrices:
A =
[-2, 4]
[1, 1]
B =
[5, -5]
[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) = (-2*5) + (4*5) = 10
C12 = (A11 * B12) + (A12 * B22) = (-2*-5) + (4*1) = 14
C21 = (A21 * B11) + (A22 * B21) = (1*5) + (1*5) = 10
C22 = (A21 * B12) + (A22 * B22) = (1*-5) + (1*1) = -4
Result Ma... |
Calculate the mean, population variance, and standard deviation for the dataset: [8, 17, 11, 20, 10] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (8 + 17 + 11 + 20 + 10) / 5 = 66 / 5 = 13.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-5.2, 3.8, -2.2, 6.8, -3.2]
Squared Differences: [27.04, 14.44, 4.84, 46.24, 10.24... |
Compute the indefinite integral of f(x) = 5x^5. | To find the indefinite integral ∫ 5x^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: 5 / 6 = 5/6.
3. Combine and add the constant of integration C.
Final Answer: ∫ 5x^5 dx = 5/6x^6 + C |
Multiply the following 2x2 matrices:
A =
[5, 0]
[-5, -2]
B =
[-4, 2]
[-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) = (5*-4) + (0*-1) = -20
C12 = (A11 * B12) + (A12 * B22) = (5*2) + (0*-3) = 10
C21 = (A21 * B11) + (A22 * B21) = (-5*-4) + (-2*-1) = 22
C22 = (A21 * B12) + (A22 * B22) = (-5*2) + (-2*-3) = -4
Re... |
Find the roots of the quadratic equation: 2x^2 + 1x + 1 = 0 | 1. Identify coefficients: a=2, b=1, c=1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1^2 - 4(2)(1) = 1 - 8 = -7.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-1 ± √7i) / 4.
Final Answer: x = -0.25 ± 0.6614378277661477i |
Calculate the mean, population variance, and standard deviation for the dataset: [12, 19, 20, 3, 3] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (12 + 19 + 20 + 3 + 3) / 5 = 57 / 5 = 11.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [0.6, 7.6, 8.6, -8.4, -8.4]
Squared Differences: [0.36, 57.76, 73.96, 70.56, 70.56]
... |
Calculate the mean, population variance, and standard deviation for the dataset: [5, 6, 20, 17, 7] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (5 + 6 + 20 + 17 + 7) / 5 = 55 / 5 = 11.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-6.0, -5.0, 9.0, 6.0, -4.0]
Squared Differences: [36.0, 25.0, 81.0, 36.0, 16.0]
S... |
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) |
Add the fractions 5/7 and 10/5. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 7 and 5, which is 35.
2. Convert 5/7 by multiplying top and bottom by 5: 25/35.
3. Convert 10/5 by multiplying top and bottom by 7: 70/35.
4. Add the numerators: 25 + 70 = 95.
5. The resulting fraction is 95/35.
6. Simplify by di... |
Multiply the following 2x2 matrices:
A =
[-3, 5]
[0, 1]
B =
[3, -5]
[-2, 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*3) + (5*-2) = -19
C12 = (A11 * B12) + (A12 * B22) = (-3*-5) + (5*5) = 40
C21 = (A21 * B11) + (A22 * B21) = (0*3) + (1*-2) = -2
C22 = (A21 * B12) + (A22 * B22) = (0*-5) + (1*5) = 5
Result ... |
Compute the quotient and remainder of 4365 divided by 94. Show the long division steps. | To divide 4365 by 94:
1. Determine how many times 94 goes into the leading digits of 4365.
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: 4365 / 94 = 46 with a remainder of 41.
Proof: (46 * 94) + 41 ... |
Multiply the following 2x2 matrices:
A =
[5, 2]
[0, 4]
B =
[-1, 1]
[-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) = (5*-1) + (2*-5) = -15
C12 = (A11 * B12) + (A12 * B22) = (5*1) + (2*-5) = -5
C21 = (A21 * B11) + (A22 * B21) = (0*-1) + (4*-5) = -20
C22 = (A21 * B12) + (A22 * B22) = (0*1) + (4*-5) = -20
Resu... |
Evaluate the logarithm: log_5(125) | To evaluate log_5(125), we ask the question: '5 raised to what power equals 125?'
Let x be the unknown power: 5^x = 125
Since 5^3 = 125, it follows that x = 3.
Final Answer: 3 |
Find the derivative of f(x) = (2x + 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 = 2x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 2x + 5. Its derivative is h'(x) = 2.
4. Multiply g'(u) by h'(x): f'(x) = 3(2x + 5)^2 * 2 = 6(2x +... |
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 +... |
Find the roots of the quadratic equation: 6x^2 + -8x + -4 = 0 | 1. Identify coefficients: a=6, b=-8, c=-4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 64 - -96 = 160.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (8 ± √160) / 12.
5. Calculate roots: x1 = 1.7207592200561266, x2 = -0.38742588672279316.
Final... |
Compute the quotient and remainder of 33042 divided by 10. Show the long division steps. | To divide 33042 by 10:
1. Determine how many times 10 goes into the leading digits of 33042.
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: 33042 / 10 = 3304 with a remainder of 2.
Proof: (3304 * 10)... |
Evaluate the logarithm: log_2(8) | To evaluate log_2(8), we ask the question: '2 raised to what power equals 8?'
Let x be the unknown power: 2^x = 8
Since 2^3 = 8, it follows that x = 3.
Final Answer: 3 |
Find the roots of the quadratic equation: 2x^2 + -6x + 10 = 0 | 1. Identify coefficients: a=2, b=-6, c=10.
2. Calculate the discriminant: Δ = b^2 - 4ac = -6^2 - 4(2)(10) = 36 - 80 = -44.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (6 ± √44i) / 4.
Final Answer: x = 1.5 ± 1.6583123951777i |
Add the fractions 9/2 and 7/3. 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 9/2 by multiplying top and bottom by 3: 27/6.
3. Convert 7/3 by multiplying top and bottom by 2: 14/6.
4. Add the numerators: 27 + 14 = 41.
5. The resulting fraction is 41/6.
Final Answer: 41/6 |
Find the roots of the quadratic equation: 10x^2 + -5x + 0 = 0 | 1. Identify coefficients: a=10, b=-5, c=0.
2. Calculate the discriminant: Δ = b^2 - 4ac = 25 - 0 = 25.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (5 ± √25) / 20.
5. Calculate roots: x1 = 0.5, x2 = 0.0.
Final Answer: x = 0.5, 0.0 |
Compute the indefinite integral of f(x) = 6x^5. | To find the indefinite integral ∫ 6x^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: 6 / 6 = 1.
3. Combine and add the constant of integration C.
Final Answer: ∫ 6x^5 dx = 1x^6 + C |
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 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 |
Multiply the following 2x2 matrices:
A =
[-4, 0]
[-5, -1]
B =
[1, -5]
[1, 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) = (-4*1) + (0*1) = -4
C12 = (A11 * B12) + (A12 * B22) = (-4*-5) + (0*2) = 20
C21 = (A21 * B11) + (A22 * B21) = (-5*1) + (-1*1) = -6
C22 = (A21 * B12) + (A22 * B22) = (-5*-5) + (-1*2) = 23
Resul... |
Calculate the mean, population variance, and standard deviation for the dataset: [6, 4, 19, 17, 13] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (6 + 4 + 19 + 17 + 13) / 5 = 59 / 5 = 11.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-5.8, -7.8, 7.2, 5.2, 1.2]
Squared Differences: [33.64, 60.84, 51.84, 27.04, 1.44]
... |
Compute the quotient and remainder of 40047 divided by 70. Show the long division steps. | To divide 40047 by 70:
1. Determine how many times 70 goes into the leading digits of 40047.
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: 40047 / 70 = 572 with a remainder of 7.
Proof: (572 * 70) +... |
Compute the quotient and remainder of 12347 divided by 30. Show the long division steps. | To divide 12347 by 30:
1. Determine how many times 30 goes into the leading digits of 12347.
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: 12347 / 30 = 411 with a remainder of 17.
Proof: (411 * 30) ... |
Find the derivative of f(x) = (4x + 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 = 4x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 5. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 5)^2 * 4 = 12(4x ... |
Evaluate cos(30°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(30°) 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) |
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 |
Calculate the mean, population variance, and standard deviation for the dataset: [15, 16, 4, 16, 9] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (15 + 16 + 4 + 16 + 9) / 5 = 60 / 5 = 12.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [3.0, 4.0, -8.0, 4.0, -3.0]
Squared Differences: [9.0, 16.0, 64.0, 16.0, 9.0]
Sum... |
Calculate the mean, population variance, and standard deviation for the dataset: [16, 8, 3, 14, 3] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (16 + 8 + 3 + 14 + 3) / 5 = 44 / 5 = 8.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [7.2, -0.8, -5.8, 5.2, -5.8]
Squared Differences: [51.84, 0.64, 33.64, 27.04, 33.64]
... |
Compute the quotient and remainder of 55099 divided by 37. Show the long division steps. | To divide 55099 by 37:
1. Determine how many times 37 goes into the leading digits of 55099.
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: 55099 / 37 = 1489 with a remainder of 6.
Proof: (1489 * 37)... |
Compute the quotient and remainder of 54996 divided by 51. Show the long division steps. | To divide 54996 by 51:
1. Determine how many times 51 goes into the leading digits of 54996.
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: 54996 / 51 = 1078 with a remainder of 18.
Proof: (1078 * 51... |
Multiply the following 2x2 matrices:
A =
[-3, -4]
[4, 3]
B =
[3, -2]
[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) = (-3*3) + (-4*4) = -25
C12 = (A11 * B12) + (A12 * B22) = (-3*-2) + (-4*1) = 2
C21 = (A21 * B11) + (A22 * B21) = (4*3) + (3*4) = 24
C22 = (A21 * B12) + (A22 * B22) = (4*-2) + (3*1) = -5
Result ... |
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... |
Find the derivative of f(x) = (2x + 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 = 2x + 3.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 2x + 3. Its derivative is h'(x) = 2.
4. Multiply g'(u) by h'(x): f'(x) = 3(2x + 3)^2 * 2 = 6(2x +... |
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 12/14 and 15/5. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 14 and 5, which is 70.
2. Convert 12/14 by multiplying top and bottom by 5: 60/70.
3. Convert 15/5 by multiplying top and bottom by 14: 210/70.
4. Add the numerators: 60 + 210 = 270.
5. The resulting fraction is 270/70.
6. Simpli... |
Calculate the mean, population variance, and standard deviation for the dataset: [6, 3, 10, 11, 17] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (6 + 3 + 10 + 11 + 17) / 5 = 47 / 5 = 9.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-3.4, -6.4, 0.6, 1.6, 7.6]
Squared Differences: [11.56, 40.96, 0.36, 2.56, 57.76]
... |
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) |
Compute the indefinite integral of f(x) = 5x^4. | To find the indefinite integral ∫ 5x^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: 5 / 5 = 1.
3. Combine and add the constant of integration C.
Final Answer: ∫ 5x^4 dx = 1x^5 + C |
Compute the quotient and remainder of 26890 divided by 87. Show the long division steps. | To divide 26890 by 87:
1. Determine how many times 87 goes into the leading digits of 26890.
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: 26890 / 87 = 309 with a remainder of 7.
Proof: (309 * 87) +... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.