prompt stringlengths 32 115 | completion stringlengths 167 566 |
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Multiply the following 2x2 matrices:
A =
[-1, -4]
[-5, -3]
B =
[-2, -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) = (-1*-2) + (-4*-4) = 18
C12 = (A11 * B12) + (A12 * B22) = (-1*-5) + (-4*-3) = 17
C21 = (A21 * B11) + (A22 * B21) = (-5*-2) + (-3*-4) = 22
C22 = (A21 * B12) + (A22 * B22) = (-5*-5) + (-3*-3) = 3... |
Multiply the following 2x2 matrices:
A =
[4, 0]
[-4, 0]
B =
[0, -2]
[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*0) + (0*1) = 0
C12 = (A11 * B12) + (A12 * B22) = (4*-2) + (0*-5) = -8
C21 = (A21 * B11) + (A22 * B21) = (-4*0) + (0*1) = 0
C22 = (A21 * B12) + (A22 * B22) = (-4*-2) + (0*-5) = 8
Result Mat... |
In a right-angled triangle, the lengths of the two legs are a = 13 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: 13^2 + 10^2 = c^2
2. Calculate the squares: 169 + 100 = c^2
3. Sum the squares: 269 = c^2
4. Take the square root of both sides: c = √269
Final A... |
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 |
Compute the quotient and remainder of 79223 divided by 9. Show the long division steps. | To divide 79223 by 9:
1. Determine how many times 9 goes into the leading digits of 79223.
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: 79223 / 9 = 8802 with a remainder of 5.
Proof: (8802 * 9) + 5... |
Find the roots of the quadratic equation: 10x^2 + 6x + -4 = 0 | 1. Identify coefficients: a=10, b=6, c=-4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 36 - -160 = 196.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-6 ± √196) / 20.
5. Calculate roots: x1 = 0.4, x2 = -1.0.
Final Answer: x = 0.4, -1.0 |
Find the roots of the quadratic equation: 6x^2 + -9x + 6 = 0 | 1. Identify coefficients: a=6, b=-9, c=6.
2. Calculate the discriminant: Δ = b^2 - 4ac = -9^2 - 4(6)(6) = 81 - 144 = -63.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (9 ± √63i) / 12.
Final Answer: x = 0.75 ± 0.6614378277661477i |
Find the roots of the quadratic equation: 3x^2 + 1x + 5 = 0 | 1. Identify coefficients: a=3, b=1, c=5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1^2 - 4(3)(5) = 1 - 60 = -59.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-1 ± √59i) / 6.
Final Answer: x = -0.16666666666666666 ± 1.2801909579781012i |
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) = 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 |
Compute the indefinite integral of f(x) = 3x^1. | To find the indefinite integral ∫ 3x^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: 3 / 2 = 3/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 3x^1 dx = 3/2x^2 + C |
Compute the quotient and remainder of 71433 divided by 10. Show the long division steps. | To divide 71433 by 10:
1. Determine how many times 10 goes into the leading digits of 71433.
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: 71433 / 10 = 7143 with a remainder of 3.
Proof: (7143 * 10)... |
Add the fractions 4/6 and 5/13. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 6 and 13, which is 78.
2. Convert 4/6 by multiplying top and bottom by 13: 52/78.
3. Convert 5/13 by multiplying top and bottom by 6: 30/78.
4. Add the numerators: 52 + 30 = 82.
5. The resulting fraction is 82/78.
6. Simplify by ... |
Evaluate the logarithm: log_3(243) | To evaluate log_3(243), we ask the question: '3 raised to what power equals 243?'
Let x be the unknown power: 3^x = 243
Since 3^5 = 243, it follows that x = 5.
Final Answer: 5 |
Compute the indefinite integral of f(x) = 2x^1. | To find the indefinite integral ∫ 2x^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: 2 / 2 = 1.
3. Combine and add the constant of integration C.
Final Answer: ∫ 2x^1 dx = 1x^2 + C |
Add the fractions 8/11 and 9/13. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 11 and 13, which is 143.
2. Convert 8/11 by multiplying top and bottom by 13: 104/143.
3. Convert 9/13 by multiplying top and bottom by 11: 99/143.
4. Add the numerators: 104 + 99 = 203.
5. The resulting fraction is 203/143.
Fina... |
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 |
Add the fractions 6/5 and 2/3. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 5 and 3, which is 15.
2. Convert 6/5 by multiplying top and bottom by 3: 18/15.
3. Convert 2/3 by multiplying top and bottom by 5: 10/15.
4. Add the numerators: 18 + 10 = 28.
5. The resulting fraction is 28/15.
Final Answer: 28/1... |
Calculate the mean, population variance, and standard deviation for the dataset: [8, 10, 4, 11, 9] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (8 + 10 + 4 + 11 + 9) / 5 = 42 / 5 = 8.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-0.4, 1.6, -4.4, 2.6, 0.6]
Squared Differences: [0.16, 2.56, 19.36, 6.76, 0.36]
Su... |
Add the fractions 6/13 and 8/4. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 13 and 4, which is 52.
2. Convert 6/13 by multiplying top and bottom by 4: 24/52.
3. Convert 8/4 by multiplying top and bottom by 13: 104/52.
4. Add the numerators: 24 + 104 = 128.
5. The resulting fraction is 128/52.
6. Simplify... |
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 +... |
Compute the quotient and remainder of 61082 divided by 93. Show the long division steps. | To divide 61082 by 93:
1. Determine how many times 93 goes into the leading digits of 61082.
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: 61082 / 93 = 656 with a remainder of 74.
Proof: (656 * 93) ... |
Find the roots of the quadratic equation: 8x^2 + -2x + 3 = 0 | 1. Identify coefficients: a=8, b=-2, c=3.
2. Calculate the discriminant: Δ = b^2 - 4ac = -2^2 - 4(8)(3) = 4 - 96 = -92.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (2 ± √92i) / 16.
Final Answer: x = 0.125 ± 0.5994789404140899i |
Find the roots of the quadratic equation: 7x^2 + -5x + 6 = 0 | 1. Identify coefficients: a=7, b=-5, c=6.
2. Calculate the discriminant: Δ = b^2 - 4ac = -5^2 - 4(7)(6) = 25 - 168 = -143.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (5 ± √143i) / 14.
Final Answer: x = 0.35714285714285715 ± 0.8541614816500999i |
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 |
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... |
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 +... |
Calculate the mean, population variance, and standard deviation for the dataset: [14, 2, 6, 19, 8] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (14 + 2 + 6 + 19 + 8) / 5 = 49 / 5 = 9.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [4.2, -7.8, -3.8, 9.2, -1.8]
Squared Differences: [17.64, 60.84, 14.44, 84.64, 3.24]
... |
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 12/14 and 6/8. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 14 and 8, which is 56.
2. Convert 12/14 by multiplying top and bottom by 4: 48/56.
3. Convert 6/8 by multiplying top and bottom by 7: 42/56.
4. Add the numerators: 48 + 42 = 90.
5. The resulting fraction is 90/56.
6. Simplify by ... |
Compute the indefinite integral of f(x) = 1x^4. | To find the indefinite integral ∫ 1x^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: 1 / 5 = 1/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 1x^4 dx = 1/5x^5 + C |
In a right-angled triangle, the lengths of the two legs are a = 5 and b = 4. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 5^2 + 4^2 = c^2
2. Calculate the squares: 25 + 16 = c^2
3. Sum the squares: 41 = c^2
4. Take the square root of both sides: c = √41
Final Answer:... |
Compute the quotient and remainder of 79703 divided by 56. Show the long division steps. | To divide 79703 by 56:
1. Determine how many times 56 goes into the leading digits of 79703.
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: 79703 / 56 = 1423 with a remainder of 15.
Proof: (1423 * 56... |
Evaluate the logarithm: log_2(16) | To evaluate log_2(16), we ask the question: '2 raised to what power equals 16?'
Let x be the unknown power: 2^x = 16
Since 2^4 = 16, it follows that x = 4.
Final Answer: 4 |
Find the roots of the quadratic equation: 6x^2 + -6x + -3 = 0 | 1. Identify coefficients: a=6, b=-6, c=-3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 36 - -72 = 108.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (6 ± √108) / 12.
5. Calculate roots: x1 = 1.3660254037844386, x2 = -0.36602540378443865.
Final... |
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 +... |
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 ... |
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 cos(45°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(45°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.7071 (exact form depends on root simplification if applicable) |
Find the roots of the quadratic equation: 4x^2 + -2x + 10 = 0 | 1. Identify coefficients: a=4, b=-2, c=10.
2. Calculate the discriminant: Δ = b^2 - 4ac = -2^2 - 4(4)(10) = 4 - 160 = -156.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (2 ± √156i) / 8.
Final Answer: x = 0.25 ± 1.5612494995995996i |
Find the roots of the quadratic equation: 2x^2 + 3x + 6 = 0 | 1. Identify coefficients: a=2, b=3, c=6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 3^2 - 4(2)(6) = 9 - 48 = -39.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-3 ± √39i) / 4.
Final Answer: x = -0.75 ± 1.5612494995995996i |
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 +... |
In a right-angled triangle, the lengths of the two legs are a = 13 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: 13^2 + 5^2 = c^2
2. Calculate the squares: 169 + 25 = c^2
3. Sum the squares: 194 = c^2
4. Take the square root of both sides: c = √194
Final Ans... |
Compute the quotient and remainder of 32047 divided by 11. Show the long division steps. | To divide 32047 by 11:
1. Determine how many times 11 goes into the leading digits of 32047.
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: 32047 / 11 = 2913 with a remainder of 4.
Proof: (2913 * 11)... |
Find the roots of the quadratic equation: 8x^2 + 6x + 5 = 0 | 1. Identify coefficients: a=8, b=6, c=5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 6^2 - 4(8)(5) = 36 - 160 = -124.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-6 ± √124i) / 16.
Final Answer: x = -0.375 ± 0.6959705453537527i |
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) |
Add the fractions 5/11 and 5/8. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 11 and 8, which is 88.
2. Convert 5/11 by multiplying top and bottom by 8: 40/88.
3. Convert 5/8 by multiplying top and bottom by 11: 55/88.
4. Add the numerators: 40 + 55 = 95.
5. The resulting fraction is 95/88.
Final Answer: 9... |
Calculate the mean, population variance, and standard deviation for the dataset: [10, 16, 20, 8, 1] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (10 + 16 + 20 + 8 + 1) / 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, 9.0, -3.0, -10.0]
Squared Differences: [1.0, 25.0, 81.0, 9.0, 100.0]
... |
Evaluate the logarithm: log_4(256) | To evaluate log_4(256), we ask the question: '4 raised to what power equals 256?'
Let x be the unknown power: 4^x = 256
Since 4^4 = 256, it follows that x = 4.
Final Answer: 4 |
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 =
[-5, 2]
[-4, -5]
B =
[5, 1]
[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) = (-5*5) + (2*2) = -21
C12 = (A11 * B12) + (A12 * B22) = (-5*1) + (2*-5) = -15
C21 = (A21 * B11) + (A22 * B21) = (-4*5) + (-5*2) = -30
C22 = (A21 * B12) + (A22 * B22) = (-4*1) + (-5*-5) = 21
Re... |
Multiply the following 2x2 matrices:
A =
[4, 1]
[0, 2]
B =
[-5, -3]
[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*-5) + (1*0) = -20
C12 = (A11 * B12) + (A12 * B22) = (4*-3) + (1*0) = -12
C21 = (A21 * B11) + (A22 * B21) = (0*-5) + (2*0) = 0
C22 = (A21 * B12) + (A22 * B22) = (0*-3) + (2*0) = 0
Result Ma... |
Compute the indefinite integral of f(x) = 6x^3. | To find the indefinite integral ∫ 6x^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: 6 / 4 = 3/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 6x^3 dx = 3/2x^4 + C |
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 |
Find the roots of the quadratic equation: 7x^2 + 6x + 4 = 0 | 1. Identify coefficients: a=7, b=6, c=4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 6^2 - 4(7)(4) = 36 - 112 = -76.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-6 ± √76i) / 14.
Final Answer: x = -0.42857142857142855 ± 0.6226998490772392i |
Multiply the following 2x2 matrices:
A =
[2, -2]
[-2, 5]
B =
[1, 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) = (2*1) + (-2*4) = -6
C12 = (A11 * B12) + (A12 * B22) = (2*2) + (-2*-1) = 6
C21 = (A21 * B11) + (A22 * B21) = (-2*1) + (5*4) = 18
C22 = (A21 * B12) + (A22 * B22) = (-2*2) + (5*-1) = -9
Result M... |
Calculate the mean, population variance, and standard deviation for the dataset: [9, 6, 10, 3, 3] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (9 + 6 + 10 + 3 + 3) / 5 = 31 / 5 = 6.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [2.8, -0.2, 3.8, -3.2, -3.2]
Squared Differences: [7.84, 0.04, 14.44, 10.24, 10.24]
... |
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 |
Add the fractions 12/4 and 3/8. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 4 and 8, which is 8.
2. Convert 12/4 by multiplying top and bottom by 2: 24/8.
3. Convert 3/8 by multiplying top and bottom by 1: 3/8.
4. Add the numerators: 24 + 3 = 27.
5. The resulting fraction is 27/8.
Final Answer: 27/8 |
Compute the indefinite integral of f(x) = 9x^4. | To find the indefinite integral ∫ 9x^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: 9 / 5 = 9/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 9x^4 dx = 9/5x^5 + C |
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 |
Compute the quotient and remainder of 6877 divided by 32. Show the long division steps. | To divide 6877 by 32:
1. Determine how many times 32 goes into the leading digits of 6877.
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: 6877 / 32 = 214 with a remainder of 29.
Proof: (214 * 32) + 2... |
Calculate the mean, population variance, and standard deviation for the dataset: [12, 9, 6, 8, 18] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (12 + 9 + 6 + 8 + 18) / 5 = 53 / 5 = 10.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [1.4, -1.6, -4.6, -2.6, 7.4]
Squared Differences: [1.96, 2.56, 21.16, 6.76, 54.76]
... |
Calculate the mean, population variance, and standard deviation for the dataset: [16, 16, 8, 14, 13] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (16 + 16 + 8 + 14 + 13) / 5 = 67 / 5 = 13.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [2.6, 2.6, -5.4, 0.6, -0.4]
Squared Differences: [6.76, 6.76, 29.16, 0.36, 0.16]
... |
Calculate the mean, population variance, and standard deviation for the dataset: [14, 3, 15, 16, 9] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (14 + 3 + 15 + 16 + 9) / 5 = 57 / 5 = 11.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [2.6, -8.4, 3.6, 4.6, -2.4]
Squared Differences: [6.76, 70.56, 12.96, 21.16, 5.76]
... |
Add the fractions 9/8 and 15/13. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 8 and 13, which is 104.
2. Convert 9/8 by multiplying top and bottom by 13: 117/104.
3. Convert 15/13 by multiplying top and bottom by 8: 120/104.
4. Add the numerators: 117 + 120 = 237.
5. The resulting fraction is 237/104.
Fina... |
Calculate the mean, population variance, and standard deviation for the dataset: [19, 2, 20, 11, 4] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (19 + 2 + 20 + 11 + 4) / 5 = 56 / 5 = 11.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [7.8, -9.2, 8.8, -0.2, -7.2]
Squared Differences: [60.84, 84.64, 77.44, 0.04, 51.84]... |
Compute the quotient and remainder of 95799 divided by 65. Show the long division steps. | To divide 95799 by 65:
1. Determine how many times 65 goes into the leading digits of 95799.
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: 95799 / 65 = 1473 with a remainder of 54.
Proof: (1473 * 65... |
Multiply the following 2x2 matrices:
A =
[1, 0]
[5, 2]
B =
[2, -3]
[-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) = (1*2) + (0*-4) = 2
C12 = (A11 * B12) + (A12 * B22) = (1*-3) + (0*3) = -3
C21 = (A21 * B11) + (A22 * B21) = (5*2) + (2*-4) = 2
C22 = (A21 * B12) + (A22 * B22) = (5*-3) + (2*3) = -9
Result Matr... |
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 |
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 |
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 |
In a right-angled triangle, the lengths of the two legs are a = 13 and b = 20. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 13^2 + 20^2 = c^2
2. Calculate the squares: 169 + 400 = c^2
3. Sum the squares: 569 = c^2
4. Take the square root of both sides: c = √569
Final A... |
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 |
Add the fractions 10/11 and 5/7. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 11 and 7, which is 77.
2. Convert 10/11 by multiplying top and bottom by 7: 70/77.
3. Convert 5/7 by multiplying top and bottom by 11: 55/77.
4. Add the numerators: 70 + 55 = 125.
5. The resulting fraction is 125/77.
Final Answer... |
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 ... |
Compute the quotient and remainder of 27314 divided by 79. Show the long division steps. | To divide 27314 by 79:
1. Determine how many times 79 goes into the leading digits of 27314.
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: 27314 / 79 = 345 with a remainder of 59.
Proof: (345 * 79) ... |
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 +... |
Multiply the following 2x2 matrices:
A =
[1, 5]
[3, 1]
B =
[4, 3]
[-1, 0] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (1*4) + (5*-1) = -1
C12 = (A11 * B12) + (A12 * B22) = (1*3) + (5*0) = 3
C21 = (A21 * B11) + (A22 * B21) = (3*4) + (1*-1) = 11
C22 = (A21 * B12) + (A22 * B22) = (3*3) + (1*0) = 9
Result Matrix... |
Compute the quotient and remainder of 46012 divided by 71. Show the long division steps. | To divide 46012 by 71:
1. Determine how many times 71 goes into the leading digits of 46012.
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: 46012 / 71 = 648 with a remainder of 4.
Proof: (648 * 71) +... |
Multiply the following 2x2 matrices:
A =
[-3, 1]
[4, -4]
B =
[3, -5]
[-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) + (1*-4) = -13
C12 = (A11 * B12) + (A12 * B22) = (-3*-5) + (1*-1) = 14
C21 = (A21 * B11) + (A22 * B21) = (4*3) + (-4*-4) = 28
C22 = (A21 * B12) + (A22 * B22) = (4*-5) + (-4*-1) = -16
R... |
Calculate the mean, population variance, and standard deviation for the dataset: [16, 17, 12, 15, 2] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (16 + 17 + 12 + 15 + 2) / 5 = 62 / 5 = 12.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [3.6, 4.6, -0.4, 2.6, -10.4]
Squared Differences: [12.96, 21.16, 0.16, 6.76, 108.16... |
Find the roots of the quadratic equation: 4x^2 + 0x + -2 = 0 | 1. Identify coefficients: a=4, b=0, c=-2.
2. Calculate the discriminant: Δ = b^2 - 4ac = 0 - -32 = 32.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (0 ± √32) / 8.
5. Calculate roots: x1 = 0.7071067811865476, x2 = -0.7071067811865476.
Final Answe... |
Add the fractions 6/15 and 1/11. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 15 and 11, which is 165.
2. Convert 6/15 by multiplying top and bottom by 11: 66/165.
3. Convert 1/11 by multiplying top and bottom by 15: 15/165.
4. Add the numerators: 66 + 15 = 81.
5. The resulting fraction is 81/165.
6. Simpl... |
Add the fractions 7/8 and 2/4. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 8 and 4, which is 8.
2. Convert 7/8 by multiplying top and bottom by 1: 7/8.
3. Convert 2/4 by multiplying top and bottom by 2: 4/8.
4. Add the numerators: 7 + 4 = 11.
5. The resulting fraction is 11/8.
Final Answer: 11/8 |
Find the roots of the quadratic equation: 8x^2 + 9x + -8 = 0 | 1. Identify coefficients: a=8, b=9, c=-8.
2. Calculate the discriminant: Δ = b^2 - 4ac = 81 - -256 = 337.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-9 ± √337) / 16.
5. Calculate roots: x1 = 0.5848474844178637, x2 = -1.7098474844178637.
Final... |
Compute the indefinite integral of f(x) = 6x^3. | To find the indefinite integral ∫ 6x^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: 6 / 4 = 3/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 6x^3 dx = 3/2x^4 + C |
Multiply the following 2x2 matrices:
A =
[-2, 2]
[-2, -2]
B =
[2, -5]
[-4, 0] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-2*2) + (2*-4) = -12
C12 = (A11 * B12) + (A12 * B22) = (-2*-5) + (2*0) = 10
C21 = (A21 * B11) + (A22 * B21) = (-2*2) + (-2*-4) = 4
C22 = (A21 * B12) + (A22 * B22) = (-2*-5) + (-2*0) = 10
Res... |
Multiply the following 2x2 matrices:
A =
[-4, 2]
[3, 5]
B =
[-5, 4]
[-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) = (-4*-5) + (2*-5) = 10
C12 = (A11 * B12) + (A12 * B22) = (-4*4) + (2*-3) = -22
C21 = (A21 * B11) + (A22 * B21) = (3*-5) + (5*-5) = -40
C22 = (A21 * B12) + (A22 * B22) = (3*4) + (5*-3) = -3
Res... |
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^4. | To find the indefinite integral ∫ 6x^4 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 4 + 1 = 5.
2. Divide the coefficient by the new exponent: 6 / 5 = 6/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 6x^4 dx = 6/5x^5 + C |
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 |
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 |
Multiply the following 2x2 matrices:
A =
[3, 1]
[0, 0]
B =
[4, -4]
[3, 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) = (3*4) + (1*3) = 15
C12 = (A11 * B12) + (A12 * B22) = (3*-4) + (1*0) = -12
C21 = (A21 * B11) + (A22 * B21) = (0*4) + (0*3) = 0
C22 = (A21 * B12) + (A22 * B22) = (0*-4) + (0*0) = 0
Result Matri... |
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 |
Compute the indefinite integral of f(x) = 4x^4. | To find the indefinite integral ∫ 4x^4 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 4 + 1 = 5.
2. Divide the coefficient by the new exponent: 4 / 5 = 4/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 4x^4 dx = 4/5x^5 + C |
Evaluate the logarithm: log_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 |
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 roots of the quadratic equation: 3x^2 + 3x + -10 = 0 | 1. Identify coefficients: a=3, b=3, c=-10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 9 - -120 = 129.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-3 ± √129) / 6.
5. Calculate roots: x1 = 1.3929694486000912, x2 = -2.392969448600091.
Final A... |
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... |
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