prompt stringlengths 32 115 | completion stringlengths 167 566 |
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Multiply the following 2x2 matrices:
A =
[-2, -2]
[-3, -4]
B =
[3, 2]
[-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) = (-2*3) + (-2*-2) = -2
C12 = (A11 * B12) + (A12 * B22) = (-2*2) + (-2*2) = -8
C21 = (A21 * B11) + (A22 * B21) = (-3*3) + (-4*-2) = -1
C22 = (A21 * B12) + (A22 * B22) = (-3*2) + (-4*2) = -14
Re... |
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 |
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) |
Compute the quotient and remainder of 2501 divided by 34. Show the long division steps. | To divide 2501 by 34:
1. Determine how many times 34 goes into the leading digits of 2501.
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: 2501 / 34 = 73 with a remainder of 19.
Proof: (73 * 34) + 19 ... |
Calculate the mean, population variance, and standard deviation for the dataset: [18, 18, 12, 19, 6] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (18 + 18 + 12 + 19 + 6) / 5 = 73 / 5 = 14.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [3.4, 3.4, -2.6, 4.4, -8.6]
Squared Differences: [11.56, 11.56, 6.76, 19.36, 73.96]... |
Find the derivative of f(x) = (2x + 4)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 2x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 2x + 4. Its derivative is h'(x) = 2.
4. Multiply g'(u) by h'(x): f'(x) = 3(2x + 4)^2 * 2 = 6(2x +... |
Calculate the mean, population variance, and standard deviation for the dataset: [17, 6, 13, 20, 12] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (17 + 6 + 13 + 20 + 12) / 5 = 68 / 5 = 13.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [3.4, -7.6, -0.6, 6.4, -1.6]
Squared Differences: [11.56, 57.76, 0.36, 40.96, 2.56]... |
Calculate the mean, population variance, and standard deviation for the dataset: [10, 3, 1, 14, 18] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (10 + 3 + 1 + 14 + 18) / 5 = 46 / 5 = 9.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [0.8, -6.2, -8.2, 4.8, 8.8]
Squared Differences: [0.64, 38.44, 67.24, 23.04, 77.44]
... |
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 ... |
Find the roots of the quadratic equation: 4x^2 + -5x + 5 = 0 | 1. Identify coefficients: a=4, b=-5, c=5.
2. Calculate the discriminant: Δ = b^2 - 4ac = -5^2 - 4(4)(5) = 25 - 80 = -55.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (5 ± √55i) / 8.
Final Answer: x = 0.625 ± 0.9270248108869579i |
Add the fractions 10/12 and 6/4. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 12 and 4, which is 12.
2. Convert 10/12 by multiplying top and bottom by 1: 10/12.
3. Convert 6/4 by multiplying top and bottom by 3: 18/12.
4. Add the numerators: 10 + 18 = 28.
5. The resulting fraction is 28/12.
6. Simplify by ... |
Compute the quotient and remainder of 91125 divided by 23. Show the long division steps. | To divide 91125 by 23:
1. Determine how many times 23 goes into the leading digits of 91125.
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: 91125 / 23 = 3961 with a remainder of 22.
Proof: (3961 * 23... |
Find the roots of the quadratic equation: 8x^2 + -8x + -7 = 0 | 1. Identify coefficients: a=8, b=-8, c=-7.
2. Calculate the discriminant: Δ = b^2 - 4ac = 64 - -224 = 288.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (8 ± √288) / 16.
5. Calculate roots: x1 = 1.5606601717798212, x2 = -0.5606601717798212.
Final... |
Evaluate tan(30°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(30°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.5774 (exact form depends on root simplification if applicable) |
Evaluate sin(0°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(0°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.0000 (exact form depends on root simplification if applicable) |
Calculate the mean, population variance, and standard deviation for the dataset: [19, 19, 11, 7, 1] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (19 + 19 + 11 + 7 + 1) / 5 = 57 / 5 = 11.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [7.6, 7.6, -0.4, -4.4, -10.4]
Squared Differences: [57.76, 57.76, 0.16, 19.36, 108.1... |
Find the roots of the quadratic equation: 3x^2 + -4x + 6 = 0 | 1. Identify coefficients: a=3, b=-4, c=6.
2. Calculate the discriminant: Δ = b^2 - 4ac = -4^2 - 4(3)(6) = 16 - 72 = -56.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (4 ± √56i) / 6.
Final Answer: x = 0.6666666666666666 ± 1.247219128924647i |
Compute the quotient and remainder of 32563 divided by 20. Show the long division steps. | To divide 32563 by 20:
1. Determine how many times 20 goes into the leading digits of 32563.
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: 32563 / 20 = 1628 with a remainder of 3.
Proof: (1628 * 20)... |
In a right-angled triangle, the lengths of the two legs are a = 4 and b = 19. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 4^2 + 19^2 = c^2
2. Calculate the squares: 16 + 361 = c^2
3. Sum the squares: 377 = c^2
4. Take the square root of both sides: c = √377
Final Ans... |
Compute the quotient and remainder of 84899 divided by 87. Show the long division steps. | To divide 84899 by 87:
1. Determine how many times 87 goes into the leading digits of 84899.
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: 84899 / 87 = 975 with a remainder of 74.
Proof: (975 * 87) ... |
In a right-angled triangle, the lengths of the two legs are a = 14 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: 14^2 + 4^2 = c^2
2. Calculate the squares: 196 + 16 = 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) = (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 +... |
Calculate the mean, population variance, and standard deviation for the dataset: [5, 2, 8, 18, 3] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (5 + 2 + 8 + 18 + 3) / 5 = 36 / 5 = 7.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-2.2, -5.2, 0.8, 10.8, -4.2]
Squared Differences: [4.84, 27.04, 0.64, 116.64, 17.64]
... |
In a right-angled triangle, the lengths of the two legs are a = 15 and b = 11. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 15^2 + 11^2 = c^2
2. Calculate the squares: 225 + 121 = c^2
3. Sum the squares: 346 = c^2
4. Take the square root of both sides: c = √346
Final A... |
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) |
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 |
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 |
In a right-angled triangle, the lengths of the two legs are a = 13 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: 13^2 + 9^2 = c^2
2. Calculate the squares: 169 + 81 = c^2
3. Sum the squares: 250 = c^2
4. Take the square root of both sides: c = √250
Final Ans... |
Find the roots of the quadratic equation: 8x^2 + -4x + -1 = 0 | 1. Identify coefficients: a=8, b=-4, c=-1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 16 - -32 = 48.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (4 ± √48) / 16.
5. Calculate roots: x1 = 0.6830127018922193, x2 = -0.1830127018922193.
Final An... |
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) |
Compute the indefinite integral of f(x) = 9x^3. | To find the indefinite integral ∫ 9x^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: 9 / 4 = 9/4.
3. Combine and add the constant of integration C.
Final Answer: ∫ 9x^3 dx = 9/4x^4 + C |
Calculate the mean, population variance, and standard deviation for the dataset: [8, 11, 1, 1, 10] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (8 + 11 + 1 + 1 + 10) / 5 = 31 / 5 = 6.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [1.8, 4.8, -5.2, -5.2, 3.8]
Squared Differences: [3.24, 23.04, 27.04, 27.04, 14.44]
... |
Multiply the following 2x2 matrices:
A =
[-3, -2]
[-1, 5]
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) = (-3*-4) + (-2*-5) = 22
C12 = (A11 * B12) + (A12 * B22) = (-3*2) + (-2*-1) = -4
C21 = (A21 * B11) + (A22 * B21) = (-1*-4) + (5*-5) = -21
C22 = (A21 * B12) + (A22 * B22) = (-1*2) + (5*-1) = -7
... |
Add the fractions 3/12 and 13/11. 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 3/12 by multiplying top and bottom by 11: 33/132.
3. Convert 13/11 by multiplying top and bottom by 12: 156/132.
4. Add the numerators: 33 + 156 = 189.
5. The resulting fraction is 189/132.
6. ... |
Evaluate the logarithm: log_4(16) | To evaluate log_4(16), we ask the question: '4 raised to what power equals 16?'
Let x be the unknown power: 4^x = 16
Since 4^2 = 16, it follows that x = 2.
Final Answer: 2 |
Find the roots of the quadratic equation: 10x^2 + -10x + 4 = 0 | 1. Identify coefficients: a=10, b=-10, c=4.
2. Calculate the discriminant: Δ = b^2 - 4ac = -10^2 - 4(10)(4) = 100 - 160 = -60.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (10 ± √60i) / 20.
Final Answer: x = 0.5 ± 0.3872983346207417i |
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: [15, 11, 12, 14, 16] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (15 + 11 + 12 + 14 + 16) / 5 = 68 / 5 = 13.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [1.4, -2.6, -1.6, 0.4, 2.4]
Squared Differences: [1.96, 6.76, 2.56, 0.16, 5.76]
... |
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 ... |
Find the roots of the quadratic equation: 8x^2 + 5x + 8 = 0 | 1. Identify coefficients: a=8, b=5, c=8.
2. Calculate the discriminant: Δ = b^2 - 4ac = 5^2 - 4(8)(8) = 25 - 256 = -231.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-5 ± √231i) / 16.
Final Answer: x = -0.3125 ± 0.9499177595981665i |
In a right-angled triangle, the lengths of the two legs are a = 19 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: 19^2 + 9^2 = c^2
2. Calculate the squares: 361 + 81 = c^2
3. Sum the squares: 442 = c^2
4. Take the square root of both sides: c = √442
Final Ans... |
Evaluate sin(0°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(0°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.0000 (exact form depends on root simplification if applicable) |
Calculate the mean, population variance, and standard deviation for the dataset: [13, 3, 1, 19, 19] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (13 + 3 + 1 + 19 + 19) / 5 = 55 / 5 = 11.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [2.0, -8.0, -10.0, 8.0, 8.0]
Squared Differences: [4.0, 64.0, 100.0, 64.0, 64.0]
... |
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) |
Calculate the mean, population variance, and standard deviation for the dataset: [9, 20, 16, 18, 6] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (9 + 20 + 16 + 18 + 6) / 5 = 69 / 5 = 13.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-4.8, 6.2, 2.2, 4.2, -7.8]
Squared Differences: [23.04, 38.44, 4.84, 17.64, 60.84]
... |
Compute the indefinite integral of f(x) = 10x^3. | To find the indefinite integral ∫ 10x^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: 10 / 4 = 5/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 10x^3 dx = 5/2x^4 + C |
Compute the quotient and remainder of 63203 divided by 60. Show the long division steps. | To divide 63203 by 60:
1. Determine how many times 60 goes into the leading digits of 63203.
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: 63203 / 60 = 1053 with a remainder of 23.
Proof: (1053 * 60... |
Multiply the following 2x2 matrices:
A =
[1, -5]
[-4, 1]
B =
[2, -5]
[3, 2] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (1*2) + (-5*3) = -13
C12 = (A11 * B12) + (A12 * B22) = (1*-5) + (-5*2) = -15
C21 = (A21 * B11) + (A22 * B21) = (-4*2) + (1*3) = -5
C22 = (A21 * B12) + (A22 * B22) = (-4*-5) + (1*2) = 22
Resul... |
Evaluate the logarithm: log_4(1024) | To evaluate log_4(1024), we ask the question: '4 raised to what power equals 1024?'
Let x be the unknown power: 4^x = 1024
Since 4^5 = 1024, it follows that x = 5.
Final Answer: 5 |
Find the roots of the quadratic equation: 6x^2 + -2x + 0 = 0 | 1. Identify coefficients: a=6, b=-2, c=0.
2. Calculate the discriminant: Δ = b^2 - 4ac = 4 - 0 = 4.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (2 ± √4) / 12.
5. Calculate roots: x1 = 0.3333333333333333, x2 = 0.0.
Final Answer: x = 0.3333333333... |
Multiply the following 2x2 matrices:
A =
[-3, -3]
[5, 0]
B =
[-4, 1]
[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) = (-3*-4) + (-3*5) = -3
C12 = (A11 * B12) + (A12 * B22) = (-3*1) + (-3*1) = -6
C21 = (A21 * B11) + (A22 * B21) = (5*-4) + (0*5) = -20
C22 = (A21 * B12) + (A22 * B22) = (5*1) + (0*1) = 5
Result ... |
Compute the indefinite integral of f(x) = 4x^5. | To find the indefinite integral ∫ 4x^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: 4 / 6 = 2/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 4x^5 dx = 2/3x^6 + C |
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 75770 divided by 25. Show the long division steps. | To divide 75770 by 25:
1. Determine how many times 25 goes into the leading digits of 75770.
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: 75770 / 25 = 3030 with a remainder of 20.
Proof: (3030 * 25... |
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 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 |
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 |
Find the roots of the quadratic equation: 3x^2 + 4x + 7 = 0 | 1. Identify coefficients: a=3, b=4, c=7.
2. Calculate the discriminant: Δ = b^2 - 4ac = 4^2 - 4(3)(7) = 16 - 84 = -68.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-4 ± √68i) / 6.
Final Answer: x = -0.6666666666666666 ± 1.3743685418725535i |
Compute the indefinite integral of f(x) = 8x^4. | To find the indefinite integral ∫ 8x^4 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 4 + 1 = 5.
2. Divide the coefficient by the new exponent: 8 / 5 = 8/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 8x^4 dx = 8/5x^5 + C |
In a right-angled triangle, the lengths of the two legs are a = 7 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: 7^2 + 10^2 = c^2
2. Calculate the squares: 49 + 100 = c^2
3. Sum the squares: 149 = c^2
4. Take the square root of both sides: c = √149
Final Ans... |
Find the roots of the quadratic equation: 7x^2 + -9x + -4 = 0 | 1. Identify coefficients: a=7, b=-9, c=-4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 81 - -112 = 193.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (9 ± √193) / 14.
5. Calculate roots: x1 = 1.6351745706749858, x2 = -0.3494602849607003.
Final... |
In a right-angled triangle, the lengths of the two legs are a = 15 and b = 13. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 15^2 + 13^2 = c^2
2. Calculate the squares: 225 + 169 = c^2
3. Sum the squares: 394 = c^2
4. Take the square root of both sides: c = √394
Final A... |
Find the roots of the quadratic equation: 8x^2 + 4x + -8 = 0 | 1. Identify coefficients: a=8, b=4, c=-8.
2. Calculate the discriminant: Δ = b^2 - 4ac = 16 - -256 = 272.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-4 ± √272) / 16.
5. Calculate roots: x1 = 0.7807764064044151, x2 = -1.2807764064044151.
Final... |
Add the fractions 8/6 and 1/10. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 6 and 10, which is 30.
2. Convert 8/6 by multiplying top and bottom by 5: 40/30.
3. Convert 1/10 by multiplying top and bottom by 3: 3/30.
4. Add the numerators: 40 + 3 = 43.
5. The resulting fraction is 43/30.
Final Answer: 43/3... |
Compute the quotient and remainder of 87354 divided by 87. Show the long division steps. | To divide 87354 by 87:
1. Determine how many times 87 goes into the leading digits of 87354.
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: 87354 / 87 = 1004 with a remainder of 6.
Proof: (1004 * 87)... |
Compute the quotient and remainder of 69743 divided by 27. Show the long division steps. | To divide 69743 by 27:
1. Determine how many times 27 goes into the leading digits of 69743.
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: 69743 / 27 = 2583 with a remainder of 2.
Proof: (2583 * 27)... |
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 +... |
Multiply the following 2x2 matrices:
A =
[1, 3]
[3, -2]
B =
[-2, -5]
[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) = (1*-2) + (3*2) = 4
C12 = (A11 * B12) + (A12 * B22) = (1*-5) + (3*4) = 7
C21 = (A21 * B11) + (A22 * B21) = (3*-2) + (-2*2) = -10
C22 = (A21 * B12) + (A22 * B22) = (3*-5) + (-2*4) = -23
Result ... |
Find the roots of the quadratic equation: 8x^2 + -1x + -5 = 0 | 1. Identify coefficients: a=8, b=-1, c=-5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1 - -160 = 161.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (1 ± √161) / 16.
5. Calculate roots: x1 = 0.855536096278095, x2 = -0.730536096278095.
Final An... |
In a right-angled triangle, the lengths of the two legs are a = 14 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: 14^2 + 16^2 = c^2
2. Calculate the squares: 196 + 256 = c^2
3. Sum the squares: 452 = c^2
4. Take the square root of both sides: c = √452
Final A... |
Find the roots of the quadratic equation: 7x^2 + 9x + -8 = 0 | 1. Identify coefficients: a=7, b=9, c=-8.
2. Calculate the discriminant: Δ = b^2 - 4ac = 81 - -224 = 305.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-9 ± √305) / 14.
5. Calculate roots: x1 = 0.6045892283266413, x2 = -1.890303514040927.
Final ... |
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 ... |
Compute the quotient and remainder of 88567 divided by 51. Show the long division steps. | To divide 88567 by 51:
1. Determine how many times 51 goes into the leading digits of 88567.
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: 88567 / 51 = 1736 with a remainder of 31.
Proof: (1736 * 51... |
Multiply the following 2x2 matrices:
A =
[3, 0]
[4, -1]
B =
[4, 0]
[2, -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*4) + (0*2) = 12
C12 = (A11 * B12) + (A12 * B22) = (3*0) + (0*-1) = 0
C21 = (A21 * B11) + (A22 * B21) = (4*4) + (-1*2) = 14
C22 = (A21 * B12) + (A22 * B22) = (4*0) + (-1*-1) = 1
Result Matr... |
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 + 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 ... |
In a right-angled triangle, the lengths of the two legs are a = 9 and b = 17. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 9^2 + 17^2 = c^2
2. Calculate the squares: 81 + 289 = c^2
3. Sum the squares: 370 = c^2
4. Take the square root of both sides: c = √370
Final Ans... |
Find the derivative of f(x) = (5x + 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 = 5x + 2.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 5x + 2. Its derivative is h'(x) = 5.
4. Multiply g'(u) by h'(x): f'(x) = 3(5x + 2)^2 * 5 = 15(5x ... |
Multiply the following 2x2 matrices:
A =
[-2, -4]
[1, -5]
B =
[-1, 3]
[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*-1) + (-4*1) = -2
C12 = (A11 * B12) + (A12 * B22) = (-2*3) + (-4*5) = -26
C21 = (A21 * B11) + (A22 * B21) = (1*-1) + (-5*1) = -6
C22 = (A21 * B12) + (A22 * B22) = (1*3) + (-5*5) = -22
Res... |
Find the roots of the quadratic equation: 10x^2 + -8x + 4 = 0 | 1. Identify coefficients: a=10, b=-8, c=4.
2. Calculate the discriminant: Δ = b^2 - 4ac = -8^2 - 4(10)(4) = 64 - 160 = -96.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (8 ± √96i) / 20.
Final Answer: x = 0.4 ± 0.4898979485566356i |
In a right-angled triangle, the lengths of the two legs are a = 9 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: 9^2 + 9^2 = c^2
2. Calculate the squares: 81 + 81 = c^2
3. Sum the squares: 162 = c^2
4. Take the square root of both sides: c = √162
Final Answe... |
Compute the indefinite integral of f(x) = 7x^4. | To find the indefinite integral ∫ 7x^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: 7 / 5 = 7/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 7x^4 dx = 7/5x^5 + C |
Find the roots of the quadratic equation: 5x^2 + 7x + -4 = 0 | 1. Identify coefficients: a=5, b=7, c=-4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 49 - -80 = 129.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-7 ± √129) / 10.
5. Calculate roots: x1 = 0.43578166916005473, x2 = -1.8357816691600548.
Final... |
Calculate the mean, population variance, and standard deviation for the dataset: [14, 19, 10, 17, 17] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (14 + 19 + 10 + 17 + 17) / 5 = 77 / 5 = 15.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-1.4, 3.6, -5.4, 1.6, 1.6]
Squared Differences: [1.96, 12.96, 29.16, 2.56, 2.56]
... |
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 |
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 |
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_5(625) | To evaluate log_5(625), we ask the question: '5 raised to what power equals 625?'
Let x be the unknown power: 5^x = 625
Since 5^4 = 625, it follows that x = 4.
Final Answer: 4 |
Find the roots of the quadratic equation: 4x^2 + 4x + -3 = 0 | 1. Identify coefficients: a=4, b=4, c=-3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 16 - -48 = 64.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-4 ± √64) / 8.
5. Calculate roots: x1 = 0.5, x2 = -1.5.
Final Answer: x = 0.5, -1.5 |
Compute the quotient and remainder of 92189 divided by 44. Show the long division steps. | To divide 92189 by 44:
1. Determine how many times 44 goes into the leading digits of 92189.
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: 92189 / 44 = 2095 with a remainder of 9.
Proof: (2095 * 44)... |
Find the roots of the quadratic equation: 6x^2 + 9x + 9 = 0 | 1. Identify coefficients: a=6, b=9, c=9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 9^2 - 4(6)(9) = 81 - 216 = -135.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-9 ± √135i) / 12.
Final Answer: x = -0.75 ± 0.9682458365518541i |
Multiply the following 2x2 matrices:
A =
[-4, 4]
[-4, 0]
B =
[1, -5]
[1, -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) + (4*1) = 0
C12 = (A11 * B12) + (A12 * B22) = (-4*-5) + (4*-4) = 4
C21 = (A21 * B11) + (A22 * B21) = (-4*1) + (0*1) = -4
C22 = (A21 * B12) + (A22 * B22) = (-4*-5) + (0*-4) = 20
Result ... |
Evaluate the logarithm: log_4(1024) | To evaluate log_4(1024), we ask the question: '4 raised to what power equals 1024?'
Let x be the unknown power: 4^x = 1024
Since 4^5 = 1024, it follows that x = 5.
Final Answer: 5 |
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 ... |
Calculate the mean, population variance, and standard deviation for the dataset: [2, 19, 7, 10, 10] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (2 + 19 + 7 + 10 + 10) / 5 = 48 / 5 = 9.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-7.6, 9.4, -2.6, 0.4, 0.4]
Squared Differences: [57.76, 88.36, 6.76, 0.16, 0.16]
... |
Compute the quotient and remainder of 51361 divided by 7. Show the long division steps. | To divide 51361 by 7:
1. Determine how many times 7 goes into the leading digits of 51361.
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: 51361 / 7 = 7337 with a remainder of 2.
Proof: (7337 * 7) + 2... |
Compute the quotient and remainder of 91232 divided by 53. Show the long division steps. | To divide 91232 by 53:
1. Determine how many times 53 goes into the leading digits of 91232.
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: 91232 / 53 = 1721 with a remainder of 19.
Proof: (1721 * 53... |
Find the roots of the quadratic equation: 6x^2 + 7x + 3 = 0 | 1. Identify coefficients: a=6, b=7, c=3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 7^2 - 4(6)(3) = 49 - 72 = -23.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-7 ± √23i) / 12.
Final Answer: x = -0.5833333333333334 ± 0.3996526269427266i |
In a right-angled triangle, the lengths of the two legs are a = 14 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: 14^2 + 15^2 = c^2
2. Calculate the squares: 196 + 225 = c^2
3. Sum the squares: 421 = c^2
4. Take the square root of both sides: c = √421
Final A... |
Compute the indefinite integral of f(x) = 1x^3. | To find the indefinite integral ∫ 1x^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: 1 / 4 = 1/4.
3. Combine and add the constant of integration C.
Final Answer: ∫ 1x^3 dx = 1/4x^4 + C |
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