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Find the roots of the quadratic equation: 4x^2 + -9x + 4 = 0 | 1. Identify coefficients: a=4, b=-9, c=4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 81 - 64 = 17.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (9 ± √17) / 8.
5. Calculate roots: x1 = 1.6403882032022077, x2 = 0.6096117967977924.
Final Answer... |
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 ... |
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 |
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 |
Find the roots of the quadratic equation: 10x^2 + 10x + 3 = 0 | 1. Identify coefficients: a=10, b=10, c=3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 10^2 - 4(10)(3) = 100 - 120 = -20.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-10 ± √20i) / 20.
Final Answer: x = -0.5 ± 0.223606797749979i |
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 22134 divided by 45. Show the long division steps. | To divide 22134 by 45:
1. Determine how many times 45 goes into the leading digits of 22134.
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: 22134 / 45 = 491 with a remainder of 39.
Proof: (491 * 45) ... |
Compute the quotient and remainder of 98070 divided by 28. Show the long division steps. | To divide 98070 by 28:
1. Determine how many times 28 goes into the leading digits of 98070.
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: 98070 / 28 = 3502 with a remainder of 14.
Proof: (3502 * 28... |
Multiply the following 2x2 matrices:
A =
[1, 5]
[0, 3]
B =
[-1, -1]
[-4, 5] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (1*-1) + (5*-4) = -21
C12 = (A11 * B12) + (A12 * B22) = (1*-1) + (5*5) = 24
C21 = (A21 * B11) + (A22 * B21) = (0*-1) + (3*-4) = -12
C22 = (A21 * B12) + (A22 * B22) = (0*-1) + (3*5) = 15
Resul... |
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) |
Add the fractions 1/4 and 4/8. 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 1/4 by multiplying top and bottom by 2: 2/8.
3. Convert 4/8 by multiplying top and bottom by 1: 4/8.
4. Add the numerators: 2 + 4 = 6.
5. The resulting fraction is 6/8.
6. Simplify by dividing nume... |
Add the fractions 8/13 and 2/8. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 13 and 8, which is 104.
2. Convert 8/13 by multiplying top and bottom by 8: 64/104.
3. Convert 2/8 by multiplying top and bottom by 13: 26/104.
4. Add the numerators: 64 + 26 = 90.
5. The resulting fraction is 90/104.
6. Simplify... |
Compute the quotient and remainder of 76237 divided by 77. Show the long division steps. | To divide 76237 by 77:
1. Determine how many times 77 goes into the leading digits of 76237.
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: 76237 / 77 = 990 with a remainder of 7.
Proof: (990 * 77) +... |
Evaluate the logarithm: log_2(16) | To evaluate log_2(16), we ask the question: '2 raised to what power equals 16?'
Let x be the unknown power: 2^x = 16
Since 2^4 = 16, it follows that x = 4.
Final Answer: 4 |
Calculate the mean, population variance, and standard deviation for the dataset: [11, 3, 6, 4, 11] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (11 + 3 + 6 + 4 + 11) / 5 = 35 / 5 = 7.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [4.0, -4.0, -1.0, -3.0, 4.0]
Squared Differences: [16.0, 16.0, 1.0, 9.0, 16.0]
Sum ... |
Calculate the mean, population variance, and standard deviation for the dataset: [15, 9, 14, 3, 12] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (15 + 9 + 14 + 3 + 12) / 5 = 53 / 5 = 10.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [4.4, -1.6, 3.4, -7.6, 1.4]
Squared Differences: [19.36, 2.56, 11.56, 57.76, 1.96]
... |
Calculate the mean, population variance, and standard deviation for the dataset: [6, 11, 19, 13, 10] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (6 + 11 + 19 + 13 + 10) / 5 = 59 / 5 = 11.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-5.8, -0.8, 7.2, 1.2, -1.8]
Squared Differences: [33.64, 0.64, 51.84, 1.44, 3.24]
... |
Compute the indefinite integral of f(x) = 7x^3. | To find the indefinite integral ∫ 7x^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: 7 / 4 = 7/4.
3. Combine and add the constant of integration C.
Final Answer: ∫ 7x^3 dx = 7/4x^4 + C |
Multiply the following 2x2 matrices:
A =
[4, -4]
[-3, 4]
B =
[3, 0]
[-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*3) + (-4*-1) = 16
C12 = (A11 * B12) + (A12 * B22) = (4*0) + (-4*-4) = 16
C21 = (A21 * B11) + (A22 * B21) = (-3*3) + (4*-1) = -13
C22 = (A21 * B12) + (A22 * B22) = (-3*0) + (4*-4) = -16
Res... |
Add the fractions 15/5 and 7/13. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 5 and 13, which is 65.
2. Convert 15/5 by multiplying top and bottom by 13: 195/65.
3. Convert 7/13 by multiplying top and bottom by 5: 35/65.
4. Add the numerators: 195 + 35 = 230.
5. The resulting fraction is 230/65.
6. Simplif... |
Compute the indefinite integral of f(x) = 5x^2. | To find the indefinite integral ∫ 5x^2 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 2 + 1 = 3.
2. Divide the coefficient by the new exponent: 5 / 3 = 5/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 5x^2 dx = 5/3x^3 + C |
Compute the indefinite integral of f(x) = 7x^2. | To find the indefinite integral ∫ 7x^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: 7 / 3 = 7/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 7x^2 dx = 7/3x^3 + C |
Compute the quotient and remainder of 47155 divided by 41. Show the long division steps. | To divide 47155 by 41:
1. Determine how many times 41 goes into the leading digits of 47155.
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: 47155 / 41 = 1150 with a remainder of 5.
Proof: (1150 * 41)... |
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... |
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 +... |
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 |
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 39857 divided by 85. Show the long division steps. | To divide 39857 by 85:
1. Determine how many times 85 goes into the leading digits of 39857.
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: 39857 / 85 = 468 with a remainder of 77.
Proof: (468 * 85) ... |
Calculate the mean, population variance, and standard deviation for the dataset: [14, 2, 1, 12, 18] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (14 + 2 + 1 + 12 + 18) / 5 = 47 / 5 = 9.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [4.6, -7.4, -8.4, 2.6, 8.6]
Squared Differences: [21.16, 54.76, 70.56, 6.76, 73.96]
... |
Add the fractions 2/7 and 14/3. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 7 and 3, which is 21.
2. Convert 2/7 by multiplying top and bottom by 3: 6/21.
3. Convert 14/3 by multiplying top and bottom by 7: 98/21.
4. Add the numerators: 6 + 98 = 104.
5. The resulting fraction is 104/21.
Final Answer: 104... |
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 ... |
Calculate the mean, population variance, and standard deviation for the dataset: [5, 2, 12, 11, 10] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (5 + 2 + 12 + 11 + 10) / 5 = 40 / 5 = 8.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-3.0, -6.0, 4.0, 3.0, 2.0]
Squared Differences: [9.0, 36.0, 16.0, 9.0, 4.0]
Sum o... |
Find the derivative of f(x) = (4x + 3)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 4x + 3.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 3. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 3)^2 * 4 = 12(4x ... |
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 |
Find the roots of the quadratic equation: 6x^2 + 6x + -7 = 0 | 1. Identify coefficients: a=6, b=6, c=-7.
2. Calculate the discriminant: Δ = b^2 - 4ac = 36 - -168 = 204.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-6 ± √204) / 12.
5. Calculate roots: x1 = 0.6902380714238084, x2 = -1.6902380714238083.
Final... |
Find the derivative of f(x) = (2x + 3)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 2x + 3.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 2x + 3. Its derivative is h'(x) = 2.
4. Multiply g'(u) by h'(x): f'(x) = 3(2x + 3)^2 * 2 = 6(2x +... |
Compute the quotient and remainder of 7597 divided by 51. Show the long division steps. | To divide 7597 by 51:
1. Determine how many times 51 goes into the leading digits of 7597.
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: 7597 / 51 = 148 with a remainder of 49.
Proof: (148 * 51) + 4... |
Multiply the following 2x2 matrices:
A =
[-4, -1]
[1, -4]
B =
[2, 3]
[0, -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*2) + (-1*0) = -8
C12 = (A11 * B12) + (A12 * B22) = (-4*3) + (-1*-3) = -9
C21 = (A21 * B11) + (A22 * B21) = (1*2) + (-4*0) = 2
C22 = (A21 * B12) + (A22 * B22) = (1*3) + (-4*-3) = 15
Result... |
Multiply the following 2x2 matrices:
A =
[-2, 4]
[3, -1]
B =
[1, 2]
[3, -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) + (4*3) = 10
C12 = (A11 * B12) + (A12 * B22) = (-2*2) + (4*-1) = -8
C21 = (A21 * B11) + (A22 * B21) = (3*1) + (-1*3) = 0
C22 = (A21 * B12) + (A22 * B22) = (3*2) + (-1*-1) = 7
Result Ma... |
Find the roots of the quadratic equation: 8x^2 + 4x + 3 = 0 | 1. Identify coefficients: a=8, b=4, c=3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 4^2 - 4(8)(3) = 16 - 96 = -80.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-4 ± √80i) / 16.
Final Answer: x = -0.25 ± 0.5590169943749475i |
Find the derivative of f(x) = (3x + 2)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 3x + 2.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 2. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 2)^2 * 3 = 9(3x +... |
Find the derivative of f(x) = (4x + 5)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 4x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 5. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 5)^2 * 4 = 12(4x ... |
Evaluate 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 |
Add the fractions 8/10 and 7/3. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 10 and 3, which is 30.
2. Convert 8/10 by multiplying top and bottom by 3: 24/30.
3. Convert 7/3 by multiplying top and bottom by 10: 70/30.
4. Add the numerators: 24 + 70 = 94.
5. The resulting fraction is 94/30.
6. Simplify by ... |
Multiply the following 2x2 matrices:
A =
[2, 0]
[-2, 0]
B =
[4, 1]
[2, -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*4) + (0*2) = 8
C12 = (A11 * B12) + (A12 * B22) = (2*1) + (0*-3) = 2
C21 = (A21 * B11) + (A22 * B21) = (-2*4) + (0*2) = -8
C22 = (A21 * B12) + (A22 * B22) = (-2*1) + (0*-3) = -2
Result Matr... |
In a right-angled triangle, the lengths of the two legs are a = 12 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: 12^2 + 17^2 = c^2
2. Calculate the squares: 144 + 289 = c^2
3. Sum the squares: 433 = c^2
4. Take the square root of both sides: c = √433
Final A... |
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 |
Find the roots of the quadratic equation: 1x^2 + -7x + -3 = 0 | 1. Identify coefficients: a=1, b=-7, c=-3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 49 - -12 = 61.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (7 ± √61) / 2.
5. Calculate roots: x1 = 7.405124837953327, x2 = -0.405124837953327.
Final Answe... |
In a right-angled triangle, the lengths of the two legs are a = 6 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: 6^2 + 19^2 = c^2
2. Calculate the squares: 36 + 361 = c^2
3. Sum the squares: 397 = c^2
4. Take the square root of both sides: c = √397
Final Ans... |
Multiply the following 2x2 matrices:
A =
[-4, -5]
[5, -2]
B =
[5, -3]
[5, -1] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-4*5) + (-5*5) = -45
C12 = (A11 * B12) + (A12 * B22) = (-4*-3) + (-5*-1) = 17
C21 = (A21 * B11) + (A22 * B21) = (5*5) + (-2*5) = 15
C22 = (A21 * B12) + (A22 * B22) = (5*-3) + (-2*-1) = -13
R... |
In a right-angled triangle, the lengths of the two legs are a = 10 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: 10^2 + 11^2 = c^2
2. Calculate the squares: 100 + 121 = c^2
3. Sum the squares: 221 = c^2
4. Take the square root of both sides: c = √221
Final A... |
In a right-angled triangle, the lengths of the two legs are a = 6 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: 6^2 + 17^2 = c^2
2. Calculate the squares: 36 + 289 = c^2
3. Sum the squares: 325 = c^2
4. Take the square root of both sides: c = √325
Final Ans... |
Multiply the following 2x2 matrices:
A =
[2, -3]
[1, 4]
B =
[-3, -3]
[-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) = (2*-3) + (-3*-3) = 3
C12 = (A11 * B12) + (A12 * B22) = (2*-3) + (-3*-2) = 0
C21 = (A21 * B11) + (A22 * B21) = (1*-3) + (4*-3) = -15
C22 = (A21 * B12) + (A22 * B22) = (1*-3) + (4*-2) = -11
Res... |
Evaluate the logarithm: log_5(125) | To evaluate log_5(125), we ask the question: '5 raised to what power equals 125?'
Let x be the unknown power: 5^x = 125
Since 5^3 = 125, it follows that x = 3.
Final Answer: 3 |
Compute the quotient and remainder of 91403 divided by 93. Show the long division steps. | To divide 91403 by 93:
1. Determine how many times 93 goes into the leading digits of 91403.
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: 91403 / 93 = 982 with a remainder of 77.
Proof: (982 * 93) ... |
Calculate the mean, population variance, and standard deviation for the dataset: [18, 16, 11, 5, 1] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (18 + 16 + 11 + 5 + 1) / 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, 5.8, 0.8, -5.2, -9.2]
Squared Differences: [60.84, 33.64, 0.64, 27.04, 84.64]
... |
In a right-angled triangle, the lengths of the two legs are a = 5 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: 5^2 + 19^2 = c^2
2. Calculate the squares: 25 + 361 = c^2
3. Sum the squares: 386 = c^2
4. Take the square root of both sides: c = √386
Final Ans... |
Compute the quotient and remainder of 75173 divided by 17. Show the long division steps. | To divide 75173 by 17:
1. Determine how many times 17 goes into the leading digits of 75173.
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: 75173 / 17 = 4421 with a remainder of 16.
Proof: (4421 * 17... |
Find the roots of the quadratic equation: 1x^2 + 2x + -9 = 0 | 1. Identify coefficients: a=1, b=2, c=-9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 4 - -36 = 40.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-2 ± √40) / 2.
5. Calculate roots: x1 = 2.1622776601683795, x2 = -4.16227766016838.
Final Answer... |
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 roots of the quadratic equation: 2x^2 + -5x + 8 = 0 | 1. Identify coefficients: a=2, b=-5, c=8.
2. Calculate the discriminant: Δ = b^2 - 4ac = -5^2 - 4(2)(8) = 25 - 64 = -39.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (5 ± √39i) / 4.
Final Answer: x = 1.25 ± 1.5612494995995996i |
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) |
In a right-angled triangle, the lengths of the two legs are a = 10 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: 10^2 + 11^2 = c^2
2. Calculate the squares: 100 + 121 = c^2
3. Sum the squares: 221 = c^2
4. Take the square root of both sides: c = √221
Final A... |
Multiply the following 2x2 matrices:
A =
[-3, 2]
[-2, -5]
B =
[-2, 3]
[1, -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*-2) + (2*1) = 8
C12 = (A11 * B12) + (A12 * B22) = (-3*3) + (2*-1) = -11
C21 = (A21 * B11) + (A22 * B21) = (-2*-2) + (-5*1) = -1
C22 = (A21 * B12) + (A22 * B22) = (-2*3) + (-5*-1) = -1
Res... |
Find the roots of the quadratic equation: 4x^2 + -9x + -1 = 0 | 1. Identify coefficients: a=4, b=-9, c=-1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 81 - -16 = 97.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (9 ± √97) / 8.
5. Calculate roots: x1 = 2.356107225224513, x2 = -0.10610722522451299.
Final Ans... |
Calculate the mean, population variance, and standard deviation for the dataset: [14, 17, 13, 6, 6] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (14 + 17 + 13 + 6 + 6) / 5 = 56 / 5 = 11.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [2.8, 5.8, 1.8, -5.2, -5.2]
Squared Differences: [7.84, 33.64, 3.24, 27.04, 27.04]
... |
Find the roots of the quadratic equation: 3x^2 + -9x + -5 = 0 | 1. Identify coefficients: a=3, b=-9, c=-5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 81 - -60 = 141.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (9 ± √141) / 6.
5. Calculate roots: x1 = 3.4790570145063193, x2 = -0.4790570145063195.
Final A... |
Find the roots of the quadratic equation: 8x^2 + -1x + 9 = 0 | 1. Identify coefficients: a=8, b=-1, c=9.
2. Calculate the discriminant: Δ = b^2 - 4ac = -1^2 - 4(8)(9) = 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) / 16.
Final Answer: x = 0.0625 ± 1.0588171466310885i |
Find the derivative of f(x) = (3x + 4)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 3x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 4. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 4)^2 * 3 = 9(3x +... |
Find the roots of the quadratic equation: 5x^2 + -10x + 9 = 0 | 1. Identify coefficients: a=5, b=-10, c=9.
2. Calculate the discriminant: Δ = b^2 - 4ac = -10^2 - 4(5)(9) = 100 - 180 = -80.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (10 ± √80i) / 10.
Final Answer: x = 1.0 ± 0.894427190999916i |
Compute the indefinite integral of f(x) = 10x^2. | To find the indefinite integral ∫ 10x^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: 10 / 3 = 10/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 10x^2 dx = 10/3x^3 + C |
Add the fractions 2/5 and 4/12. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 5 and 12, which is 60.
2. Convert 2/5 by multiplying top and bottom by 12: 24/60.
3. Convert 4/12 by multiplying top and bottom by 5: 20/60.
4. Add the numerators: 24 + 20 = 44.
5. The resulting fraction is 44/60.
6. Simplify by ... |
Calculate the mean, population variance, and standard deviation for the dataset: [4, 10, 3, 6, 19] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (4 + 10 + 3 + 6 + 19) / 5 = 42 / 5 = 8.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-4.4, 1.6, -5.4, -2.4, 10.6]
Squared Differences: [19.36, 2.56, 29.16, 5.76, 112.36]
... |
Calculate the mean, population variance, and standard deviation for the dataset: [20, 3, 9, 18, 7] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (20 + 3 + 9 + 18 + 7) / 5 = 57 / 5 = 11.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [8.6, -8.4, -2.4, 6.6, -4.4]
Squared Differences: [73.96, 70.56, 5.76, 43.56, 19.36]
... |
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 tan(45°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(45°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 1.0000 (exact form depends on root simplification if applicable) |
Find the derivative of f(x) = (4x + 2)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 4x + 2.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 2. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 2)^2 * 4 = 12(4x ... |
Find the roots of the quadratic equation: 5x^2 + -5x + -5 = 0 | 1. Identify coefficients: a=5, b=-5, c=-5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 25 - -100 = 125.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (5 ± √125) / 10.
5. Calculate roots: x1 = 1.618033988749895, x2 = -0.6180339887498949.
Final ... |
Compute the quotient and remainder of 33146 divided by 19. Show the long division steps. | To divide 33146 by 19:
1. Determine how many times 19 goes into the leading digits of 33146.
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: 33146 / 19 = 1744 with a remainder of 10.
Proof: (1744 * 19... |
Add the fractions 8/14 and 7/15. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 14 and 15, which is 210.
2. Convert 8/14 by multiplying top and bottom by 15: 120/210.
3. Convert 7/15 by multiplying top and bottom by 14: 98/210.
4. Add the numerators: 120 + 98 = 218.
5. The resulting fraction is 218/210.
6. S... |
Compute the indefinite integral of f(x) = 10x^4. | To find the indefinite integral ∫ 10x^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: 10 / 5 = 2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 10x^4 dx = 2x^5 + C |
Find the roots of the quadratic equation: 7x^2 + -8x + -9 = 0 | 1. Identify coefficients: a=7, b=-8, c=-9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 64 - -252 = 316.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (8 ± √316) / 14.
5. Calculate roots: x1 = 1.8411706310450842, x2 = -0.6983134881879413.
Final... |
Evaluate the logarithm: log_2(32) | To evaluate log_2(32), we ask the question: '2 raised to what power equals 32?'
Let x be the unknown power: 2^x = 32
Since 2^5 = 32, it follows that x = 5.
Final Answer: 5 |
Multiply the following 2x2 matrices:
A =
[-5, -4]
[-5, 0]
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) = (-5*1) + (-4*-4) = 11
C12 = (A11 * B12) + (A12 * B22) = (-5*-2) + (-4*-1) = 14
C21 = (A21 * B11) + (A22 * B21) = (-5*1) + (0*-4) = -5
C22 = (A21 * B12) + (A22 * B22) = (-5*-2) + (0*-1) = 10
R... |
Add the fractions 15/9 and 3/8. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 9 and 8, which is 72.
2. Convert 15/9 by multiplying top and bottom by 8: 120/72.
3. Convert 3/8 by multiplying top and bottom by 9: 27/72.
4. Add the numerators: 120 + 27 = 147.
5. The resulting fraction is 147/72.
6. Simplify b... |
Evaluate the logarithm: log_2(32) | To evaluate log_2(32), we ask the question: '2 raised to what power equals 32?'
Let x be the unknown power: 2^x = 32
Since 2^5 = 32, it follows that x = 5.
Final Answer: 5 |
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 ... |
Multiply the following 2x2 matrices:
A =
[-5, -2]
[-2, 2]
B =
[0, 5]
[4, 5] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-5*0) + (-2*4) = -8
C12 = (A11 * B12) + (A12 * B22) = (-5*5) + (-2*5) = -35
C21 = (A21 * B11) + (A22 * B21) = (-2*0) + (2*4) = 8
C22 = (A21 * B12) + (A22 * B22) = (-2*5) + (2*5) = 0
Result M... |
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 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 |
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 =
[-3, 5]
[5, 3]
B =
[0, 1]
[5, 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*0) + (5*5) = 25
C12 = (A11 * B12) + (A12 * B22) = (-3*1) + (5*0) = -3
C21 = (A21 * B11) + (A22 * B21) = (5*0) + (3*5) = 15
C22 = (A21 * B12) + (A22 * B22) = (5*1) + (3*0) = 5
Result Matri... |
Add the fractions 4/14 and 4/6. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 14 and 6, which is 42.
2. Convert 4/14 by multiplying top and bottom by 3: 12/42.
3. Convert 4/6 by multiplying top and bottom by 7: 28/42.
4. Add the numerators: 12 + 28 = 40.
5. The resulting fraction is 40/42.
6. Simplify by d... |
Compute the quotient and remainder of 3035 divided by 32. Show the long division steps. | To divide 3035 by 32:
1. Determine how many times 32 goes into the leading digits of 3035.
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: 3035 / 32 = 94 with a remainder of 27.
Proof: (94 * 32) + 27 ... |
Add the fractions 15/8 and 1/2. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 8 and 2, which is 8.
2. Convert 15/8 by multiplying top and bottom by 1: 15/8.
3. Convert 1/2 by multiplying top and bottom by 4: 4/8.
4. Add the numerators: 15 + 4 = 19.
5. The resulting fraction is 19/8.
Final Answer: 19/8 |
Add the fractions 8/11 and 12/6. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 11 and 6, which is 66.
2. Convert 8/11 by multiplying top and bottom by 6: 48/66.
3. Convert 12/6 by multiplying top and bottom by 11: 132/66.
4. Add the numerators: 48 + 132 = 180.
5. The resulting fraction is 180/66.
6. Simplif... |
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 |
In a right-angled triangle, the lengths of the two legs are a = 5 and b = 8. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 5^2 + 8^2 = c^2
2. Calculate the squares: 25 + 64 = c^2
3. Sum the squares: 89 = c^2
4. Take the square root of both sides: c = √89
Final Answer:... |
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) |
Add the fractions 6/2 and 4/5. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 2 and 5, which is 10.
2. Convert 6/2 by multiplying top and bottom by 5: 30/10.
3. Convert 4/5 by multiplying top and bottom by 2: 8/10.
4. Add the numerators: 30 + 8 = 38.
5. The resulting fraction is 38/10.
6. Simplify by divid... |
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