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Find the roots of the quadratic equation: 3x^2 + 5x + 10 = 0 | 1. Identify coefficients: a=3, b=5, c=10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 5^2 - 4(3)(10) = 25 - 120 = -95.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-5 ± √95i) / 6.
Final Answer: x = -0.8333333333333334 ± 1.6244657241348273i |
Compute the indefinite integral of f(x) = 2x^3. | To find the indefinite integral ∫ 2x^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: 2 / 4 = 1/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 2x^3 dx = 1/2x^4 + C |
Multiply the following 2x2 matrices:
A =
[-1, 4]
[3, 2]
B =
[-1, -5]
[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) = (-1*-1) + (4*2) = 9
C12 = (A11 * B12) + (A12 * B22) = (-1*-5) + (4*-1) = 1
C21 = (A21 * B11) + (A22 * B21) = (3*-1) + (2*2) = 1
C22 = (A21 * B12) + (A22 * B22) = (3*-5) + (2*-1) = -17
Result ... |
Add the fractions 8/4 and 15/8. 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 8/4 by multiplying top and bottom by 2: 16/8.
3. Convert 15/8 by multiplying top and bottom by 1: 15/8.
4. Add the numerators: 16 + 15 = 31.
5. The resulting fraction is 31/8.
Final Answer: 31/8 |
Find the roots of the quadratic equation: 3x^2 + 5x + 10 = 0 | 1. Identify coefficients: a=3, b=5, c=10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 5^2 - 4(3)(10) = 25 - 120 = -95.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-5 ± √95i) / 6.
Final Answer: x = -0.8333333333333334 ± 1.6244657241348273i |
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 |
Calculate the mean, population variance, and standard deviation for the dataset: [8, 14, 8, 3, 7] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (8 + 14 + 8 + 3 + 7) / 5 = 40 / 5 = 8.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [0.0, 6.0, 0.0, -5.0, -1.0]
Squared Differences: [0.0, 36.0, 0.0, 25.0, 1.0]
Sum of ... |
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) = (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 49009 divided by 14. Show the long division steps. | To divide 49009 by 14:
1. Determine how many times 14 goes into the leading digits of 49009.
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: 49009 / 14 = 3500 with a remainder of 9.
Proof: (3500 * 14)... |
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 |
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... |
Calculate the mean, population variance, and standard deviation for the dataset: [13, 17, 18, 15, 19] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (13 + 17 + 18 + 15 + 19) / 5 = 82 / 5 = 16.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-3.4, 0.6, 1.6, -1.4, 2.6]
Squared Differences: [11.56, 0.36, 2.56, 1.96, 6.76]
... |
In a right-angled triangle, the lengths of the two legs are a = 7 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: 7^2 + 12^2 = c^2
2. Calculate the squares: 49 + 144 = c^2
3. Sum the squares: 193 = c^2
4. Take the square root of both sides: c = √193
Final Ans... |
In a right-angled triangle, the lengths of the two legs are a = 20 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: 20^2 + 15^2 = c^2
2. Calculate the squares: 400 + 225 = c^2
3. Sum the squares: 625 = c^2
4. Take the square root of both sides: c = √625
Final A... |
Compute the quotient and remainder of 45250 divided by 52. Show the long division steps. | To divide 45250 by 52:
1. Determine how many times 52 goes into the leading digits of 45250.
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: 45250 / 52 = 870 with a remainder of 10.
Proof: (870 * 52) ... |
Find the roots of the quadratic equation: 3x^2 + 4x + -1 = 0 | 1. Identify coefficients: a=3, b=4, c=-1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 16 - -12 = 28.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-4 ± √28) / 6.
5. Calculate roots: x1 = 0.21525043702153024, x2 = -1.5485837703548635.
Final An... |
Multiply the following 2x2 matrices:
A =
[-5, -5]
[-1, -1]
B =
[-1, 4]
[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) = (-5*-1) + (-5*2) = -5
C12 = (A11 * B12) + (A12 * B22) = (-5*4) + (-5*1) = -25
C21 = (A21 * B11) + (A22 * B21) = (-1*-1) + (-1*2) = -1
C22 = (A21 * B12) + (A22 * B22) = (-1*4) + (-1*1) = -5
Re... |
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 derivative of f(x) = (3x + 4)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 3x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 4. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 4)^2 * 3 = 9(3x +... |
Evaluate the logarithm: log_2(8) | To evaluate log_2(8), we ask the question: '2 raised to what power equals 8?'
Let x be the unknown power: 2^x = 8
Since 2^3 = 8, it follows that x = 3.
Final Answer: 3 |
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 =
[-5, -1]
[-1, -1]
B =
[0, -4]
[-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) = (-5*0) + (-1*-3) = 3
C12 = (A11 * B12) + (A12 * B22) = (-5*-4) + (-1*2) = 18
C21 = (A21 * B11) + (A22 * B21) = (-1*0) + (-1*-3) = 3
C22 = (A21 * B12) + (A22 * B22) = (-1*-4) + (-1*2) = 2
Resu... |
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(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 |
Calculate the mean, population variance, and standard deviation for the dataset: [10, 2, 15, 4, 5] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (10 + 2 + 15 + 4 + 5) / 5 = 36 / 5 = 7.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [2.8, -5.2, 7.8, -3.2, -2.2]
Squared Differences: [7.84, 27.04, 60.84, 10.24, 4.84]
... |
Find the roots of the quadratic equation: 10x^2 + 8x + 8 = 0 | 1. Identify coefficients: a=10, b=8, c=8.
2. Calculate the discriminant: Δ = b^2 - 4ac = 8^2 - 4(10)(8) = 64 - 320 = -256.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-8 ± √256i) / 20.
Final Answer: x = -0.4 ± 0.8i |
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) = (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 ... |
Compute the quotient and remainder of 31961 divided by 96. Show the long division steps. | To divide 31961 by 96:
1. Determine how many times 96 goes into the leading digits of 31961.
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: 31961 / 96 = 332 with a remainder of 89.
Proof: (332 * 96) ... |
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) |
Calculate the mean, population variance, and standard deviation for the dataset: [18, 7, 14, 10, 2] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (18 + 7 + 14 + 10 + 2) / 5 = 51 / 5 = 10.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [7.8, -3.2, 3.8, -0.2, -8.2]
Squared Differences: [60.84, 10.24, 14.44, 0.04, 67.24]... |
In a right-angled triangle, the lengths of the two legs are a = 16 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: 16^2 + 14^2 = c^2
2. Calculate the squares: 256 + 196 = c^2
3. Sum the squares: 452 = c^2
4. Take the square root of both sides: c = √452
Final A... |
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 |
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 ... |
In a right-angled triangle, the lengths of the two legs are a = 9 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: 9^2 + 5^2 = c^2
2. Calculate the squares: 81 + 25 = c^2
3. Sum the squares: 106 = c^2
4. Take the square root of both sides: c = √106
Final Answe... |
Add the fractions 10/14 and 12/2. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 14 and 2, which is 14.
2. Convert 10/14 by multiplying top and bottom by 1: 10/14.
3. Convert 12/2 by multiplying top and bottom by 7: 84/14.
4. Add the numerators: 10 + 84 = 94.
5. The resulting fraction is 94/14.
6. Simplify by... |
Calculate the mean, population variance, and standard deviation for the dataset: [1, 16, 12, 14, 2] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (1 + 16 + 12 + 14 + 2) / 5 = 45 / 5 = 9.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-8.0, 7.0, 3.0, 5.0, -7.0]
Squared Differences: [64.0, 49.0, 9.0, 25.0, 49.0]
Sum... |
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 roots of the quadratic equation: 9x^2 + 1x + 8 = 0 | 1. Identify coefficients: a=9, b=1, c=8.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1^2 - 4(9)(8) = 1 - 288 = -287.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-1 ± √287i) / 18.
Final Answer: x = -0.05555555555555555 ± 0.941170797005412i |
Find the derivative of f(x) = (3x + 4)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 3x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 4. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 4)^2 * 3 = 9(3x +... |
In a right-angled triangle, the lengths of the two legs are a = 11 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: 11^2 + 12^2 = c^2
2. Calculate the squares: 121 + 144 = c^2
3. Sum the squares: 265 = c^2
4. Take the square root of both sides: c = √265
Final A... |
Find the roots of the quadratic equation: 5x^2 + -8x + 6 = 0 | 1. Identify coefficients: a=5, b=-8, c=6.
2. Calculate the discriminant: Δ = b^2 - 4ac = -8^2 - 4(5)(6) = 64 - 120 = -56.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (8 ± √56i) / 10.
Final Answer: x = 0.8 ± 0.7483314773547882i |
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) = 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 |
Compute the indefinite integral of f(x) = 9x^5. | To find the indefinite integral ∫ 9x^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: 9 / 6 = 3/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 9x^5 dx = 3/2x^6 + C |
Compute the indefinite integral of f(x) = 9x^5. | To find the indefinite integral ∫ 9x^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: 9 / 6 = 3/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 9x^5 dx = 3/2x^6 + C |
Add the fractions 14/13 and 2/13. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 13 and 13, which is 13.
2. Convert 14/13 by multiplying top and bottom by 1: 14/13.
3. Convert 2/13 by multiplying top and bottom by 1: 2/13.
4. Add the numerators: 14 + 2 = 16.
5. The resulting fraction is 16/13.
Final Answer: 1... |
In a right-angled triangle, the lengths of the two legs are a = 7 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: 7^2 + 17^2 = c^2
2. Calculate the squares: 49 + 289 = c^2
3. Sum the squares: 338 = c^2
4. Take the square root of both sides: c = √338
Final Ans... |
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) |
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 |
In a right-angled triangle, the lengths of the two legs are a = 11 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: 11^2 + 16^2 = c^2
2. Calculate the squares: 121 + 256 = c^2
3. Sum the squares: 377 = c^2
4. Take the square root of both sides: c = √377
Final A... |
Compute the quotient and remainder of 76544 divided by 31. Show the long division steps. | To divide 76544 by 31:
1. Determine how many times 31 goes into the leading digits of 76544.
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: 76544 / 31 = 2469 with a remainder of 5.
Proof: (2469 * 31)... |
Multiply the following 2x2 matrices:
A =
[-3, 3]
[-1, -4]
B =
[-5, -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) = (-3*-5) + (3*-5) = 0
C12 = (A11 * B12) + (A12 * B22) = (-3*-4) + (3*-2) = 6
C21 = (A21 * B11) + (A22 * B21) = (-1*-5) + (-4*-5) = 25
C22 = (A21 * B12) + (A22 * B22) = (-1*-4) + (-4*-2) = 12
R... |
Multiply the following 2x2 matrices:
A =
[-2, -5]
[1, -3]
B =
[-5, 3]
[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) = (-2*-5) + (-5*1) = 5
C12 = (A11 * B12) + (A12 * B22) = (-2*3) + (-5*-4) = 14
C21 = (A21 * B11) + (A22 * B21) = (1*-5) + (-3*1) = -8
C22 = (A21 * B12) + (A22 * B22) = (1*3) + (-3*-4) = 15
Resu... |
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 |
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 +... |
Multiply the following 2x2 matrices:
A =
[0, -2]
[2, 4]
B =
[2, 1]
[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) = (0*2) + (-2*4) = -8
C12 = (A11 * B12) + (A12 * B22) = (0*1) + (-2*1) = -2
C21 = (A21 * B11) + (A22 * B21) = (2*2) + (4*4) = 20
C22 = (A21 * B12) + (A22 * B22) = (2*1) + (4*1) = 6
Result Matri... |
Compute the quotient and remainder of 51696 divided by 8. Show the long division steps. | To divide 51696 by 8:
1. Determine how many times 8 goes into the leading digits of 51696.
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: 51696 / 8 = 6462 with a remainder of 0.
Proof: (6462 * 8) + 0... |
Add the fractions 13/4 and 5/8. 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 13/4 by multiplying top and bottom by 2: 26/8.
3. Convert 5/8 by multiplying top and bottom by 1: 5/8.
4. Add the numerators: 26 + 5 = 31.
5. The resulting fraction is 31/8.
Final Answer: 31/8 |
Find the derivative of f(x) = (3x + 4)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 3x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 4. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 4)^2 * 3 = 9(3x +... |
Find the 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 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) |
Find the roots of the quadratic equation: 3x^2 + 5x + -9 = 0 | 1. Identify coefficients: a=3, b=5, c=-9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 25 - -108 = 133.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-5 ± √133) / 6.
5. Calculate roots: x1 = 1.0887604324451328, x2 = -2.7554270991117993.
Final ... |
Evaluate the logarithm: log_2(4) | To evaluate log_2(4), we ask the question: '2 raised to what power equals 4?'
Let x be the unknown power: 2^x = 4
Since 2^2 = 4, it follows that x = 2.
Final Answer: 2 |
Find the roots of the quadratic equation: 8x^2 + 6x + -3 = 0 | 1. Identify coefficients: a=8, b=6, c=-3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 36 - -96 = 132.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-6 ± √132) / 16.
5. Calculate roots: x1 = 0.3430703308172536, x2 = -1.0930703308172536.
Final ... |
Add the fractions 5/2 and 1/4. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 2 and 4, which is 4.
2. Convert 5/2 by multiplying top and bottom by 2: 10/4.
3. Convert 1/4 by multiplying top and bottom by 1: 1/4.
4. Add the numerators: 10 + 1 = 11.
5. The resulting fraction is 11/4.
Final Answer: 11/4 |
In a right-angled triangle, the lengths of the two legs are a = 3 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: 3^2 + 9^2 = c^2
2. Calculate the squares: 9 + 81 = c^2
3. Sum the squares: 90 = c^2
4. Take the square root of both sides: c = √90
Final Answer: ... |
Find the roots of the quadratic equation: 8x^2 + -6x + -9 = 0 | 1. Identify coefficients: a=8, b=-6, c=-9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 36 - -288 = 324.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (6 ± √324) / 16.
5. Calculate roots: x1 = 1.5, x2 = -0.75.
Final Answer: x = 1.5, -0.75 |
Compute the quotient and remainder of 33976 divided by 99. Show the long division steps. | To divide 33976 by 99:
1. Determine how many times 99 goes into the leading digits of 33976.
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: 33976 / 99 = 343 with a remainder of 19.
Proof: (343 * 99) ... |
Find the roots of the quadratic equation: 4x^2 + 3x + 6 = 0 | 1. Identify coefficients: a=4, b=3, c=6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 3^2 - 4(4)(6) = 9 - 96 = -87.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-3 ± √87i) / 8.
Final Answer: x = -0.375 ± 1.165922381636102i |
Multiply the following 2x2 matrices:
A =
[-4, -2]
[-2, -1]
B =
[0, 0]
[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) = (-4*0) + (-2*4) = -8
C12 = (A11 * B12) + (A12 * B22) = (-4*0) + (-2*-1) = 2
C21 = (A21 * B11) + (A22 * B21) = (-2*0) + (-1*4) = -4
C22 = (A21 * B12) + (A22 * B22) = (-2*0) + (-1*-1) = 1
Resul... |
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 +... |
Find the roots of the quadratic equation: 4x^2 + -9x + -10 = 0 | 1. Identify coefficients: a=4, b=-9, c=-10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 81 - -160 = 241.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (9 ± √241) / 8.
5. Calculate roots: x1 = 3.0655218370325032, x2 = -0.815521837032503.
Final ... |
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 |
Find the roots of the quadratic equation: 3x^2 + 4x + 3 = 0 | 1. Identify coefficients: a=3, b=4, c=3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 4^2 - 4(3)(3) = 16 - 36 = -20.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-4 ± √20i) / 6.
Final Answer: x = -0.6666666666666666 ± 0.7453559924999299i |
Calculate the mean, population variance, and standard deviation for the dataset: [2, 16, 13, 9, 6] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (2 + 16 + 13 + 9 + 6) / 5 = 46 / 5 = 9.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-7.2, 6.8, 3.8, -0.2, -3.2]
Squared Differences: [51.84, 46.24, 14.44, 0.04, 10.24]
... |
Calculate the mean, population variance, and standard deviation for the dataset: [18, 5, 8, 4, 20] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (18 + 5 + 8 + 4 + 20) / 5 = 55 / 5 = 11.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [7.0, -6.0, -3.0, -7.0, 9.0]
Squared Differences: [49.0, 36.0, 9.0, 49.0, 81.0]
Su... |
In a right-angled triangle, the lengths of the two legs are a = 19 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: 19^2 + 18^2 = c^2
2. Calculate the squares: 361 + 324 = c^2
3. Sum the squares: 685 = c^2
4. Take the square root of both sides: c = √685
Final A... |
In a right-angled triangle, the lengths of the two legs are a = 13 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: 13^2 + 14^2 = c^2
2. Calculate the squares: 169 + 196 = c^2
3. Sum the squares: 365 = c^2
4. Take the square root of both sides: c = √365
Final A... |
Multiply the following 2x2 matrices:
A =
[3, -1]
[0, 4]
B =
[0, -2]
[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) = (3*0) + (-1*1) = -1
C12 = (A11 * B12) + (A12 * B22) = (3*-2) + (-1*-2) = -4
C21 = (A21 * B11) + (A22 * B21) = (0*0) + (4*1) = 4
C22 = (A21 * B12) + (A22 * B22) = (0*-2) + (4*-2) = -8
Result M... |
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 |
Multiply the following 2x2 matrices:
A =
[2, 1]
[2, -4]
B =
[5, -5]
[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) = (2*5) + (1*1) = 11
C12 = (A11 * B12) + (A12 * B22) = (2*-5) + (1*3) = -7
C21 = (A21 * B11) + (A22 * B21) = (2*5) + (-4*1) = 6
C22 = (A21 * B12) + (A22 * B22) = (2*-5) + (-4*3) = -22
Result Ma... |
In a right-angled triangle, the lengths of the two legs are a = 14 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: 14^2 + 6^2 = c^2
2. Calculate the squares: 196 + 36 = c^2
3. Sum the squares: 232 = c^2
4. Take the square root of both sides: c = √232
Final Ans... |
Evaluate the logarithm: log_5(25) | To evaluate log_5(25), we ask the question: '5 raised to what power equals 25?'
Let x be the unknown power: 5^x = 25
Since 5^2 = 25, it follows that x = 2.
Final Answer: 2 |
In a right-angled triangle, the lengths of the two legs are a = 20 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: 20^2 + 14^2 = c^2
2. Calculate the squares: 400 + 196 = c^2
3. Sum the squares: 596 = c^2
4. Take the square root of both sides: c = √596
Final A... |
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 |
Multiply the following 2x2 matrices:
A =
[-4, -5]
[3, 2]
B =
[-5, -5]
[-1, 2] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-4*-5) + (-5*-1) = 25
C12 = (A11 * B12) + (A12 * B22) = (-4*-5) + (-5*2) = 10
C21 = (A21 * B11) + (A22 * B21) = (3*-5) + (2*-1) = -17
C22 = (A21 * B12) + (A22 * B22) = (3*-5) + (2*2) = -11
R... |
In a right-angled triangle, the lengths of the two legs are a = 4 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: 4^2 + 16^2 = c^2
2. Calculate the squares: 16 + 256 = c^2
3. Sum the squares: 272 = c^2
4. Take the square root of both sides: c = √272
Final Ans... |
In a right-angled triangle, the lengths of the two legs are a = 20 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: 20^2 + 11^2 = c^2
2. Calculate the squares: 400 + 121 = c^2
3. Sum the squares: 521 = c^2
4. Take the square root of both sides: c = √521
Final A... |
In a right-angled triangle, the lengths of the two legs are a = 18 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: 18^2 + 5^2 = c^2
2. Calculate the squares: 324 + 25 = c^2
3. Sum the squares: 349 = c^2
4. Take the square root of both sides: c = √349
Final Ans... |
Multiply the following 2x2 matrices:
A =
[2, 5]
[-5, 0]
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) = (2*2) + (5*-4) = -16
C12 = (A11 * B12) + (A12 * B22) = (2*-3) + (5*-3) = -21
C21 = (A21 * B11) + (A22 * B21) = (-5*2) + (0*-4) = -10
C22 = (A21 * B12) + (A22 * B22) = (-5*-3) + (0*-3) = 15
Re... |
Calculate the mean, population variance, and standard deviation for the dataset: [14, 17, 17, 8, 1] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (14 + 17 + 17 + 8 + 1) / 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, 5.6, 5.6, -3.4, -10.4]
Squared Differences: [6.76, 31.36, 31.36, 11.56, 108.16... |
In a right-angled triangle, the lengths of the two legs are a = 15 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: 15^2 + 3^2 = c^2
2. Calculate the squares: 225 + 9 = c^2
3. Sum the squares: 234 = c^2
4. Take the square root of both sides: c = √234
Final Answ... |
Compute the quotient and remainder of 49440 divided by 84. Show the long division steps. | To divide 49440 by 84:
1. Determine how many times 84 goes into the leading digits of 49440.
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: 49440 / 84 = 588 with a remainder of 48.
Proof: (588 * 84) ... |
Compute the quotient and remainder of 91374 divided by 31. Show the long division steps. | To divide 91374 by 31:
1. Determine how many times 31 goes into the leading digits of 91374.
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: 91374 / 31 = 2947 with a remainder of 17.
Proof: (2947 * 31... |
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 |
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 |
Calculate the mean, population variance, and standard deviation for the dataset: [14, 20, 18, 6, 8] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (14 + 20 + 18 + 6 + 8) / 5 = 66 / 5 = 13.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [0.8, 6.8, 4.8, -7.2, -5.2]
Squared Differences: [0.64, 46.24, 23.04, 51.84, 27.04]
... |
Compute the quotient and remainder of 58311 divided by 59. Show the long division steps. | To divide 58311 by 59:
1. Determine how many times 59 goes into the leading digits of 58311.
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: 58311 / 59 = 988 with a remainder of 19.
Proof: (988 * 59) ... |
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