prompt stringlengths 32 115 | completion stringlengths 167 566 |
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Multiply the following 2x2 matrices:
A =
[1, -4]
[-1, -5]
B =
[4, -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) = (1*4) + (-4*1) = 0
C12 = (A11 * B12) + (A12 * B22) = (1*-5) + (-4*2) = -13
C21 = (A21 * B11) + (A22 * B21) = (-1*4) + (-5*1) = -9
C22 = (A21 * B12) + (A22 * B22) = (-1*-5) + (-5*2) = -5
Resul... |
Multiply the following 2x2 matrices:
A =
[-3, -2]
[-3, 3]
B =
[1, 2]
[-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) = (-3*1) + (-2*-5) = 7
C12 = (A11 * B12) + (A12 * B22) = (-3*2) + (-2*4) = -14
C21 = (A21 * B11) + (A22 * B21) = (-3*1) + (3*-5) = -18
C22 = (A21 * B12) + (A22 * B22) = (-3*2) + (3*4) = 6
Resul... |
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 ... |
Add the fractions 8/2 and 11/14. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 2 and 14, which is 14.
2. Convert 8/2 by multiplying top and bottom by 7: 56/14.
3. Convert 11/14 by multiplying top and bottom by 1: 11/14.
4. Add the numerators: 56 + 11 = 67.
5. The resulting fraction is 67/14.
Final Answer: 6... |
Find the roots of the quadratic equation: 1x^2 + 8x + -3 = 0 | 1. Identify coefficients: a=1, b=8, c=-3.
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) / 2.
5. Calculate roots: x1 = 0.358898943540674, x2 = -8.358898943540673.
Final Answe... |
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 |
Calculate the mean, population variance, and standard deviation for the dataset: [12, 12, 8, 11, 19] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (12 + 12 + 8 + 11 + 19) / 5 = 62 / 5 = 12.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-0.4, -0.4, -4.4, -1.4, 6.6]
Squared Differences: [0.16, 0.16, 19.36, 1.96, 43.56]... |
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 quotient and remainder of 28893 divided by 41. Show the long division steps. | To divide 28893 by 41:
1. Determine how many times 41 goes into the leading digits of 28893.
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: 28893 / 41 = 704 with a remainder of 29.
Proof: (704 * 41) ... |
Calculate the mean, population variance, and standard deviation for the dataset: [13, 19, 8, 19, 18] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (13 + 19 + 8 + 19 + 18) / 5 = 77 / 5 = 15.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-2.4, 3.6, -7.4, 3.6, 2.6]
Squared Differences: [5.76, 12.96, 54.76, 12.96, 6.76]
... |
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) = (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(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 |
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 ... |
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 |
Add the fractions 15/9 and 14/13. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 9 and 13, which is 117.
2. Convert 15/9 by multiplying top and bottom by 13: 195/117.
3. Convert 14/13 by multiplying top and bottom by 9: 126/117.
4. Add the numerators: 195 + 126 = 321.
5. The resulting fraction is 321/117.
6. ... |
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 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 |
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 |
Add the fractions 15/4 and 13/4. 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 15/4 by multiplying top and bottom by 1: 15/4.
3. Convert 13/4 by multiplying top and bottom by 1: 13/4.
4. Add the numerators: 15 + 13 = 28.
5. The resulting fraction is 28/4.
6. Simplify by divid... |
In a right-angled triangle, the lengths of the two legs are a = 10 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: 10^2 + 3^2 = c^2
2. Calculate the squares: 100 + 9 = c^2
3. Sum the squares: 109 = c^2
4. Take the square root of both sides: c = √109
Final Answ... |
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 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) |
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 +... |
Compute the quotient and remainder of 22294 divided by 33. Show the long division steps. | To divide 22294 by 33:
1. Determine how many times 33 goes into the leading digits of 22294.
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: 22294 / 33 = 675 with a remainder of 19.
Proof: (675 * 33) ... |
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 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 roots of the quadratic equation: 10x^2 + -1x + -7 = 0 | 1. Identify coefficients: a=10, b=-1, c=-7.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1 - -280 = 281.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (1 ± √281) / 20.
5. Calculate roots: x1 = 0.8881527307120105, x2 = -0.7881527307120105.
Final... |
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 |
Multiply the following 2x2 matrices:
A =
[0, 5]
[-3, -4]
B =
[-5, 4]
[2, 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) = (0*-5) + (5*2) = 10
C12 = (A11 * B12) + (A12 * B22) = (0*4) + (5*0) = 0
C21 = (A21 * B11) + (A22 * B21) = (-3*-5) + (-4*2) = 7
C22 = (A21 * B12) + (A22 * B22) = (-3*4) + (-4*0) = -12
Result M... |
Multiply the following 2x2 matrices:
A =
[0, -1]
[4, 2]
B =
[2, 1]
[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) = (0*2) + (-1*0) = 0
C12 = (A11 * B12) + (A12 * B22) = (0*1) + (-1*4) = -4
C21 = (A21 * B11) + (A22 * B21) = (4*2) + (2*0) = 8
C22 = (A21 * B12) + (A22 * B22) = (4*1) + (2*4) = 12
Result Matrix... |
Calculate the mean, population variance, and standard deviation for the dataset: [12, 6, 13, 12, 8] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (12 + 6 + 13 + 12 + 8) / 5 = 51 / 5 = 10.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [1.8, -4.2, 2.8, 1.8, -2.2]
Squared Differences: [3.24, 17.64, 7.84, 3.24, 4.84]
... |
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) |
In a right-angled triangle, the lengths of the two legs are a = 20 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: 20^2 + 12^2 = c^2
2. Calculate the squares: 400 + 144 = c^2
3. Sum the squares: 544 = c^2
4. Take the square root of both sides: c = √544
Final A... |
Compute the quotient and remainder of 95976 divided by 78. Show the long division steps. | To divide 95976 by 78:
1. Determine how many times 78 goes into the leading digits of 95976.
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: 95976 / 78 = 1230 with a remainder of 36.
Proof: (1230 * 78... |
In a right-angled triangle, the lengths of the two legs are a = 6 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: 6^2 + 10^2 = c^2
2. Calculate the squares: 36 + 100 = c^2
3. Sum the squares: 136 = c^2
4. Take the square root of both sides: c = √136
Final Ans... |
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) |
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 37049 divided by 50. Show the long division steps. | To divide 37049 by 50:
1. Determine how many times 50 goes into the leading digits of 37049.
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: 37049 / 50 = 740 with a remainder of 49.
Proof: (740 * 50) ... |
Add the fractions 4/12 and 6/11. 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 4/12 by multiplying top and bottom by 11: 44/132.
3. Convert 6/11 by multiplying top and bottom by 12: 72/132.
4. Add the numerators: 44 + 72 = 116.
5. The resulting fraction is 116/132.
6. Sim... |
Add the fractions 11/3 and 6/7. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 3 and 7, which is 21.
2. Convert 11/3 by multiplying top and bottom by 7: 77/21.
3. Convert 6/7 by multiplying top and bottom by 3: 18/21.
4. Add the numerators: 77 + 18 = 95.
5. The resulting fraction is 95/21.
Final Answer: 95/... |
In a right-angled triangle, the lengths of the two legs are a = 17 and b = 5. 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 + 5^2 = c^2
2. Calculate the squares: 289 + 25 = c^2
3. Sum the squares: 314 = c^2
4. Take the square root of both sides: c = √314
Final Ans... |
Compute the quotient and remainder of 79560 divided by 53. Show the long division steps. | To divide 79560 by 53:
1. Determine how many times 53 goes into the leading digits of 79560.
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: 79560 / 53 = 1501 with a remainder of 7.
Proof: (1501 * 53)... |
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) |
Calculate the mean, population variance, and standard deviation for the dataset: [16, 3, 11, 20, 5] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (16 + 3 + 11 + 20 + 5) / 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, -8.0, 0.0, 9.0, -6.0]
Squared Differences: [25.0, 64.0, 0.0, 81.0, 36.0]
Su... |
In a right-angled triangle, the lengths of the two legs are a = 8 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: 8^2 + 14^2 = c^2
2. Calculate the squares: 64 + 196 = c^2
3. Sum the squares: 260 = c^2
4. Take the square root of both sides: c = √260
Final Ans... |
Evaluate cos(90°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(90°) 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) |
Compute the quotient and remainder of 95697 divided by 45. Show the long division steps. | To divide 95697 by 45:
1. Determine how many times 45 goes into the leading digits of 95697.
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: 95697 / 45 = 2126 with a remainder of 27.
Proof: (2126 * 45... |
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 = 5 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: 5^2 + 12^2 = c^2
2. Calculate the squares: 25 + 144 = c^2
3. Sum the squares: 169 = c^2
4. Take the square root of both sides: c = √169
Final Ans... |
Multiply the following 2x2 matrices:
A =
[-1, -2]
[2, -2]
B =
[5, 2]
[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) = (-1*5) + (-2*2) = -9
C12 = (A11 * B12) + (A12 * B22) = (-1*2) + (-2*-5) = 8
C21 = (A21 * B11) + (A22 * B21) = (2*5) + (-2*2) = 6
C22 = (A21 * B12) + (A22 * B22) = (2*2) + (-2*-5) = 14
Result ... |
Compute the quotient and remainder of 3762 divided by 66. Show the long division steps. | To divide 3762 by 66:
1. Determine how many times 66 goes into the leading digits of 3762.
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: 3762 / 66 = 57 with a remainder of 0.
Proof: (57 * 66) + 0 = ... |
Compute the indefinite integral of f(x) = 5x^2. | To find the indefinite integral ∫ 5x^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: 5 / 3 = 5/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 5x^2 dx = 5/3x^3 + C |
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, 18, 13, 18, 17] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (9 + 18 + 13 + 18 + 17) / 5 = 75 / 5 = 15.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-6.0, 3.0, -2.0, 3.0, 2.0]
Squared Differences: [36.0, 9.0, 4.0, 9.0, 4.0]
Sum ... |
Find the roots of the quadratic equation: 6x^2 + 6x + -10 = 0 | 1. Identify coefficients: a=6, b=6, c=-10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 36 - -240 = 276.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-6 ± √276) / 12.
5. Calculate roots: x1 = 0.8844373104863458, x2 = -1.8844373104863459.
Fina... |
Multiply the following 2x2 matrices:
A =
[0, 2]
[4, 3]
B =
[-4, 5]
[4, -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) = (0*-4) + (2*4) = 8
C12 = (A11 * B12) + (A12 * B22) = (0*5) + (2*-3) = -6
C21 = (A21 * B11) + (A22 * B21) = (4*-4) + (3*4) = -4
C22 = (A21 * B12) + (A22 * B22) = (4*5) + (3*-3) = 11
Result Mat... |
Find the roots of the quadratic equation: 10x^2 + 6x + 0 = 0 | 1. Identify coefficients: a=10, b=6, c=0.
2. Calculate the discriminant: Δ = b^2 - 4ac = 36 - 0 = 36.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-6 ± √36) / 20.
5. Calculate roots: x1 = 0.0, x2 = -0.6.
Final Answer: x = 0.0, -0.6 |
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:... |
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: 8x^2 + 6x + 8 = 0 | 1. Identify coefficients: a=8, b=6, c=8.
2. Calculate the discriminant: Δ = b^2 - 4ac = 6^2 - 4(8)(8) = 36 - 256 = -220.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-6 ± √220i) / 16.
Final Answer: x = -0.375 ± 0.9270248108869579i |
Find the roots of the quadratic equation: 2x^2 + 1x + 0 = 0 | 1. Identify coefficients: a=2, b=1, c=0.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1 - 0 = 1.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-1 ± √1) / 4.
5. Calculate roots: x1 = 0.0, x2 = -0.5.
Final Answer: x = 0.0, -0.5 |
In a right-angled triangle, the lengths of the two legs are a = 11 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: 11^2 + 6^2 = c^2
2. Calculate the squares: 121 + 36 = c^2
3. Sum the squares: 157 = c^2
4. Take the square root of both sides: c = √157
Final Ans... |
Compute the indefinite integral of f(x) = 10x^1. | To find the indefinite integral ∫ 10x^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: 10 / 2 = 5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 10x^1 dx = 5x^2 + C |
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) = 3x^4. | To find the indefinite integral ∫ 3x^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: 3 / 5 = 3/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 3x^4 dx = 3/5x^5 + C |
Multiply the following 2x2 matrices:
A =
[5, -4]
[5, 1]
B =
[4, 0]
[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) = (5*4) + (-4*4) = 4
C12 = (A11 * B12) + (A12 * B22) = (5*0) + (-4*-2) = 8
C21 = (A21 * B11) + (A22 * B21) = (5*4) + (1*4) = 24
C22 = (A21 * B12) + (A22 * B22) = (5*0) + (1*-2) = -2
Result Matr... |
In a right-angled triangle, the lengths of the two legs are a = 17 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: 17^2 + 14^2 = c^2
2. Calculate the squares: 289 + 196 = c^2
3. Sum the squares: 485 = c^2
4. Take the square root of both sides: c = √485
Final A... |
Add the fractions 14/3 and 3/12. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 3 and 12, which is 12.
2. Convert 14/3 by multiplying top and bottom by 4: 56/12.
3. Convert 3/12 by multiplying top and bottom by 1: 3/12.
4. Add the numerators: 56 + 3 = 59.
5. The resulting fraction is 59/12.
Final Answer: 59/... |
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 ... |
Find the roots of the quadratic equation: 6x^2 + 5x + 10 = 0 | 1. Identify coefficients: a=6, b=5, c=10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 5^2 - 4(6)(10) = 25 - 240 = -215.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-5 ± √215i) / 12.
Final Answer: x = -0.4166666666666667 ± 1.2219065248845984i |
Multiply the following 2x2 matrices:
A =
[-2, 0]
[5, 2]
B =
[5, 0]
[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) = (-2*5) + (0*5) = -10
C12 = (A11 * B12) + (A12 * B22) = (-2*0) + (0*3) = 0
C21 = (A21 * B11) + (A22 * B21) = (5*5) + (2*5) = 35
C22 = (A21 * B12) + (A22 * B22) = (5*0) + (2*3) = 6
Result Matri... |
Calculate the mean, population variance, and standard deviation for the dataset: [2, 4, 1, 1, 11] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (2 + 4 + 1 + 1 + 11) / 5 = 19 / 5 = 3.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-1.8, 0.2, -2.8, -2.8, 7.2]
Squared Differences: [3.24, 0.04, 7.84, 7.84, 51.84]
Su... |
Add the fractions 12/10 and 9/4. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 10 and 4, which is 20.
2. Convert 12/10 by multiplying top and bottom by 2: 24/20.
3. Convert 9/4 by multiplying top and bottom by 5: 45/20.
4. Add the numerators: 24 + 45 = 69.
5. The resulting fraction is 69/20.
Final Answer: 6... |
Multiply the following 2x2 matrices:
A =
[-1, 1]
[-4, -5]
B =
[-1, 0]
[-5, -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*-1) + (1*-5) = -4
C12 = (A11 * B12) + (A12 * B22) = (-1*0) + (1*-2) = -2
C21 = (A21 * B11) + (A22 * B21) = (-4*-1) + (-5*-5) = 29
C22 = (A21 * B12) + (A22 * B22) = (-4*0) + (-5*-2) = 10
R... |
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 9/6 and 12/15. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 6 and 15, which is 30.
2. Convert 9/6 by multiplying top and bottom by 5: 45/30.
3. Convert 12/15 by multiplying top and bottom by 2: 24/30.
4. Add the numerators: 45 + 24 = 69.
5. The resulting fraction is 69/30.
6. Simplify by ... |
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 |
Multiply the following 2x2 matrices:
A =
[-2, 0]
[4, 5]
B =
[3, -3]
[4, 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*3) + (0*4) = -6
C12 = (A11 * B12) + (A12 * B22) = (-2*-3) + (0*4) = 6
C21 = (A21 * B11) + (A22 * B21) = (4*3) + (5*4) = 32
C22 = (A21 * B12) + (A22 * B22) = (4*-3) + (5*4) = 8
Result Matr... |
Multiply the following 2x2 matrices:
A =
[-5, 5]
[0, 3]
B =
[-4, 4]
[-5, 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*-4) + (5*-5) = -5
C12 = (A11 * B12) + (A12 * B22) = (-5*4) + (5*2) = -10
C21 = (A21 * B11) + (A22 * B21) = (0*-4) + (3*-5) = -15
C22 = (A21 * B12) + (A22 * B22) = (0*4) + (3*2) = 6
Result... |
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 |
Multiply the following 2x2 matrices:
A =
[0, -4]
[-1, -1]
B =
[4, -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) = (0*4) + (-4*-5) = 20
C12 = (A11 * B12) + (A12 * B22) = (0*-2) + (-4*1) = -4
C21 = (A21 * B11) + (A22 * B21) = (-1*4) + (-1*-5) = 1
C22 = (A21 * B12) + (A22 * B22) = (-1*-2) + (-1*1) = 1
Resul... |
Multiply the following 2x2 matrices:
A =
[1, 5]
[-4, -5]
B =
[1, -4]
[-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) = (1*1) + (5*-2) = -9
C12 = (A11 * B12) + (A12 * B22) = (1*-4) + (5*2) = 6
C21 = (A21 * B11) + (A22 * B21) = (-4*1) + (-5*-2) = 6
C22 = (A21 * B12) + (A22 * B22) = (-4*-4) + (-5*2) = 6
Result M... |
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: 9x^2 + 6x + 6 = 0 | 1. Identify coefficients: a=9, b=6, c=6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 6^2 - 4(9)(6) = 36 - 216 = -180.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-6 ± √180i) / 18.
Final Answer: x = -0.3333333333333333 ± 0.7453559924999299i |
Calculate the mean, population variance, and standard deviation for the dataset: [5, 11, 19, 17, 15] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (5 + 11 + 19 + 17 + 15) / 5 = 67 / 5 = 13.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-8.4, -2.4, 5.6, 3.6, 1.6]
Squared Differences: [70.56, 5.76, 31.36, 12.96, 2.56]
... |
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 =
[4, 5]
[3, -3]
B =
[-1, -3]
[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) = (4*-1) + (5*1) = 1
C12 = (A11 * B12) + (A12 * B22) = (4*-3) + (5*3) = 3
C21 = (A21 * B11) + (A22 * B21) = (3*-1) + (-3*1) = -6
C22 = (A21 * B12) + (A22 * B22) = (3*-3) + (-3*3) = -18
Result M... |
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 +... |
Find the roots of the quadratic equation: 7x^2 + 2x + 8 = 0 | 1. Identify coefficients: a=7, b=2, c=8.
2. Calculate the discriminant: Δ = b^2 - 4ac = 2^2 - 4(7)(8) = 4 - 224 = -220.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-2 ± √220i) / 14.
Final Answer: x = -0.14285714285714285 ± 1.0594569267279519i |
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 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 = 4 and b = 9. 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 + 9^2 = c^2
2. Calculate the squares: 16 + 81 = c^2
3. Sum the squares: 97 = c^2
4. Take the square root of both sides: c = √97
Final Answer:... |
In a right-angled triangle, the lengths of the two legs are a = 9 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: 9^2 + 8^2 = c^2
2. Calculate the squares: 81 + 64 = c^2
3. Sum the squares: 145 = c^2
4. Take the square root of both sides: c = √145
Final Answe... |
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 =
[-5, 2]
[5, -3]
B =
[2, -3]
[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) = (-5*2) + (2*5) = 0
C12 = (A11 * B12) + (A12 * B22) = (-5*-3) + (2*-1) = 13
C21 = (A21 * B11) + (A22 * B21) = (5*2) + (-3*5) = -5
C22 = (A21 * B12) + (A22 * B22) = (5*-3) + (-3*-1) = -12
Resul... |
Multiply the following 2x2 matrices:
A =
[-5, 0]
[3, -5]
B =
[-2, -2]
[-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*-2) + (0*-4) = 10
C12 = (A11 * B12) + (A12 * B22) = (-5*-2) + (0*-5) = 10
C21 = (A21 * B11) + (A22 * B21) = (3*-2) + (-5*-4) = 14
C22 = (A21 * B12) + (A22 * B22) = (3*-2) + (-5*-5) = 19
R... |
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 +... |
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 |
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 ... |
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