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Compute the quotient and remainder of 67866 divided by 11. Show the long division steps. | To divide 67866 by 11:
1. Determine how many times 11 goes into the leading digits of 67866.
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: 67866 / 11 = 6169 with a remainder of 7.
Proof: (6169 * 11)... |
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: [9, 9, 7, 5, 17] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (9 + 9 + 7 + 5 + 17) / 5 = 47 / 5 = 9.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-0.4, -0.4, -2.4, -4.4, 7.6]
Squared Differences: [0.16, 0.16, 5.76, 19.36, 57.76]
... |
Calculate the mean, population variance, and standard deviation for the dataset: [1, 18, 11, 15, 10] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (1 + 18 + 11 + 15 + 10) / 5 = 55 / 5 = 11.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-10.0, 7.0, 0.0, 4.0, -1.0]
Squared Differences: [100.0, 49.0, 0.0, 16.0, 1.0]
... |
Add the fractions 4/6 and 12/7. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 6 and 7, which is 42.
2. Convert 4/6 by multiplying top and bottom by 7: 28/42.
3. Convert 12/7 by multiplying top and bottom by 6: 72/42.
4. Add the numerators: 28 + 72 = 100.
5. The resulting fraction is 100/42.
6. Simplify by ... |
Add the fractions 1/6 and 7/7. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 6 and 7, which is 42.
2. Convert 1/6 by multiplying top and bottom by 7: 7/42.
3. Convert 7/7 by multiplying top and bottom by 6: 42/42.
4. Add the numerators: 7 + 42 = 49.
5. The resulting fraction is 49/42.
6. Simplify by divid... |
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 |
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 ... |
Multiply the following 2x2 matrices:
A =
[-1, -2]
[0, 3]
B =
[3, -2]
[3, -3] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-1*3) + (-2*3) = -9
C12 = (A11 * B12) + (A12 * B22) = (-1*-2) + (-2*-3) = 8
C21 = (A21 * B11) + (A22 * B21) = (0*3) + (3*3) = 9
C22 = (A21 * B12) + (A22 * B22) = (0*-2) + (3*-3) = -9
Result ... |
Compute the quotient and remainder of 96276 divided by 46. Show the long division steps. | To divide 96276 by 46:
1. Determine how many times 46 goes into the leading digits of 96276.
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: 96276 / 46 = 2092 with a remainder of 44.
Proof: (2092 * 46... |
Calculate the mean, population variance, and standard deviation for the dataset: [7, 7, 11, 17, 9] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (7 + 7 + 11 + 17 + 9) / 5 = 51 / 5 = 10.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-3.2, -3.2, 0.8, 6.8, -1.2]
Squared Differences: [10.24, 10.24, 0.64, 46.24, 1.44]
... |
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: 2x^2 + -9x + -2 = 0 | 1. Identify coefficients: a=2, b=-9, c=-2.
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) / 4.
5. Calculate roots: x1 = 4.712214450449026, x2 = -0.21221445044902598.
Final Ans... |
Find the derivative of f(x) = (2x + 3)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 2x + 3.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 2x + 3. Its derivative is h'(x) = 2.
4. Multiply g'(u) by h'(x): f'(x) = 3(2x + 3)^2 * 2 = 6(2x +... |
Calculate the mean, population variance, and standard deviation for the dataset: [18, 5, 4, 7, 11] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (18 + 5 + 4 + 7 + 11) / 5 = 45 / 5 = 9.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [9.0, -4.0, -5.0, -2.0, 2.0]
Squared Differences: [81.0, 16.0, 25.0, 4.0, 4.0]
Sum ... |
Calculate the mean, population variance, and standard deviation for the dataset: [8, 15, 15, 20, 18] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (8 + 15 + 15 + 20 + 18) / 5 = 76 / 5 = 15.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-7.2, -0.2, -0.2, 4.8, 2.8]
Squared Differences: [51.84, 0.04, 0.04, 23.04, 7.84]
... |
Compute the indefinite integral of f(x) = 5x^4. | To find the indefinite integral ∫ 5x^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: 5 / 5 = 1.
3. Combine and add the constant of integration C.
Final Answer: ∫ 5x^4 dx = 1x^5 + C |
Multiply the following 2x2 matrices:
A =
[-4, 5]
[5, 2]
B =
[3, -2]
[-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) = (-4*3) + (5*-2) = -22
C12 = (A11 * B12) + (A12 * B22) = (-4*-2) + (5*4) = 28
C21 = (A21 * B11) + (A22 * B21) = (5*3) + (2*-2) = 11
C22 = (A21 * B12) + (A22 * B22) = (5*-2) + (2*4) = -2
Result... |
Add the fractions 3/4 and 15/9. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 4 and 9, which is 36.
2. Convert 3/4 by multiplying top and bottom by 9: 27/36.
3. Convert 15/9 by multiplying top and bottom by 4: 60/36.
4. Add the numerators: 27 + 60 = 87.
5. The resulting fraction is 87/36.
6. Simplify by di... |
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 |
Calculate the mean, population variance, and standard deviation for the dataset: [19, 9, 9, 19, 17] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (19 + 9 + 9 + 19 + 17) / 5 = 73 / 5 = 14.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [4.4, -5.6, -5.6, 4.4, 2.4]
Squared Differences: [19.36, 31.36, 31.36, 19.36, 5.76]
... |
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 ... |
Add the fractions 4/14 and 3/12. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 14 and 12, which is 84.
2. Convert 4/14 by multiplying top and bottom by 6: 24/84.
3. Convert 3/12 by multiplying top and bottom by 7: 21/84.
4. Add the numerators: 24 + 21 = 45.
5. The resulting fraction is 45/84.
6. Simplify by... |
Multiply the following 2x2 matrices:
A =
[-4, 3]
[5, -1]
B =
[1, -3]
[-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) = (-4*1) + (3*-2) = -10
C12 = (A11 * B12) + (A12 * B22) = (-4*-3) + (3*3) = 21
C21 = (A21 * B11) + (A22 * B21) = (5*1) + (-1*-2) = 7
C22 = (A21 * B12) + (A22 * B22) = (5*-3) + (-1*3) = -18
Resu... |
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: 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 = 4.16227766016838, x2 = -2.1622776601683795.
Final Answer... |
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 |
Find the derivative of f(x) = (3x + 3)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 3x + 3.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 3. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 3)^2 * 3 = 9(3x +... |
Compute the quotient and remainder of 80502 divided by 75. Show the long division steps. | To divide 80502 by 75:
1. Determine how many times 75 goes into the leading digits of 80502.
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: 80502 / 75 = 1073 with a remainder of 27.
Proof: (1073 * 75... |
Calculate the mean, population variance, and standard deviation for the dataset: [5, 5, 20, 16, 11] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (5 + 5 + 20 + 16 + 11) / 5 = 57 / 5 = 11.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-6.4, -6.4, 8.6, 4.6, -0.4]
Squared Differences: [40.96, 40.96, 73.96, 21.16, 0.16]... |
Multiply the following 2x2 matrices:
A =
[4, 0]
[-5, 1]
B =
[-2, -3]
[2, 5] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (4*-2) + (0*2) = -8
C12 = (A11 * B12) + (A12 * B22) = (4*-3) + (0*5) = -12
C21 = (A21 * B11) + (A22 * B21) = (-5*-2) + (1*2) = 12
C22 = (A21 * B12) + (A22 * B22) = (-5*-3) + (1*5) = 20
Result... |
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 = 16 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: 16^2 + 12^2 = c^2
2. Calculate the squares: 256 + 144 = c^2
3. Sum the squares: 400 = c^2
4. Take the square root of both sides: c = √400
Final A... |
Evaluate sin(45°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(45°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.7071 (exact form depends on root simplification if applicable) |
In a right-angled triangle, the lengths of the two legs are a = 17 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: 17^2 + 17^2 = c^2
2. Calculate the squares: 289 + 289 = c^2
3. Sum the squares: 578 = c^2
4. Take the square root of both sides: c = √578
Final A... |
Multiply the following 2x2 matrices:
A =
[4, -1]
[3, -5]
B =
[-5, 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) = (4*-5) + (-1*-2) = -18
C12 = (A11 * B12) + (A12 * B22) = (4*2) + (-1*2) = 6
C21 = (A21 * B11) + (A22 * B21) = (3*-5) + (-5*-2) = -5
C22 = (A21 * B12) + (A22 * B22) = (3*2) + (-5*2) = -4
Resul... |
Add the fractions 4/3 and 13/10. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 3 and 10, which is 30.
2. Convert 4/3 by multiplying top and bottom by 10: 40/30.
3. Convert 13/10 by multiplying top and bottom by 3: 39/30.
4. Add the numerators: 40 + 39 = 79.
5. The resulting fraction is 79/30.
Final Answer: ... |
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 |
Calculate the mean, population variance, and standard deviation for the dataset: [6, 13, 18, 17, 18] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (6 + 13 + 18 + 17 + 18) / 5 = 72 / 5 = 14.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-8.4, -1.4, 3.6, 2.6, 3.6]
Squared Differences: [70.56, 1.96, 12.96, 6.76, 12.96]
... |
Evaluate tan(90°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(90°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 16331239353195370.0000 (exact form depends on root simplification if applicable) |
Compute the indefinite integral of f(x) = 10x^5. | To find the indefinite integral ∫ 10x^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: 10 / 6 = 5/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 10x^5 dx = 5/3x^6 + C |
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 |
Compute the quotient and remainder of 79915 divided by 92. Show the long division steps. | To divide 79915 by 92:
1. Determine how many times 92 goes into the leading digits of 79915.
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: 79915 / 92 = 868 with a remainder of 59.
Proof: (868 * 92) ... |
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 |
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 +... |
Add the fractions 11/10 and 4/5. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 10 and 5, which is 10.
2. Convert 11/10 by multiplying top and bottom by 1: 11/10.
3. Convert 4/5 by multiplying top and bottom by 2: 8/10.
4. Add the numerators: 11 + 8 = 19.
5. The resulting fraction is 19/10.
Final Answer: 19/... |
Add the fractions 12/13 and 4/12. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 13 and 12, which is 156.
2. Convert 12/13 by multiplying top and bottom by 12: 144/156.
3. Convert 4/12 by multiplying top and bottom by 13: 52/156.
4. Add the numerators: 144 + 52 = 196.
5. The resulting fraction is 196/156.
6. ... |
Calculate the mean, population variance, and standard deviation for the dataset: [18, 13, 9, 10, 9] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (18 + 13 + 9 + 10 + 9) / 5 = 59 / 5 = 11.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [6.2, 1.2, -2.8, -1.8, -2.8]
Squared Differences: [38.44, 1.44, 7.84, 3.24, 7.84]
... |
Evaluate sin(90°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(90°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 1.0000 (exact form depends on root simplification if applicable) |
Find the roots of the quadratic equation: 8x^2 + 10x + -3 = 0 | 1. Identify coefficients: a=8, b=10, c=-3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 100 - -96 = 196.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-10 ± √196) / 16.
5. Calculate roots: x1 = 0.25, x2 = -1.5.
Final Answer: x = 0.25, -1.5 |
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) |
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 |
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 |
Calculate the mean, population variance, and standard deviation for the dataset: [4, 8, 18, 1, 19] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (4 + 8 + 18 + 1 + 19) / 5 = 50 / 5 = 10.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-6.0, -2.0, 8.0, -9.0, 9.0]
Squared Differences: [36.0, 4.0, 64.0, 81.0, 81.0]
Su... |
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) |
Add the fractions 10/8 and 1/15. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 8 and 15, which is 120.
2. Convert 10/8 by multiplying top and bottom by 15: 150/120.
3. Convert 1/15 by multiplying top and bottom by 8: 8/120.
4. Add the numerators: 150 + 8 = 158.
5. The resulting fraction is 158/120.
6. Simpl... |
Compute the quotient and remainder of 10875 divided by 9. Show the long division steps. | To divide 10875 by 9:
1. Determine how many times 9 goes into the leading digits of 10875.
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: 10875 / 9 = 1208 with a remainder of 3.
Proof: (1208 * 9) + 3... |
Add the fractions 7/13 and 15/9. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 13 and 9, which is 117.
2. Convert 7/13 by multiplying top and bottom by 9: 63/117.
3. Convert 15/9 by multiplying top and bottom by 13: 195/117.
4. Add the numerators: 63 + 195 = 258.
5. The resulting fraction is 258/117.
6. Sim... |
Multiply the following 2x2 matrices:
A =
[3, 0]
[-2, -1]
B =
[2, 0]
[-3, 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) = (3*2) + (0*-3) = 6
C12 = (A11 * B12) + (A12 * B22) = (3*0) + (0*3) = 0
C21 = (A21 * B11) + (A22 * B21) = (-2*2) + (-1*-3) = -1
C22 = (A21 * B12) + (A22 * B22) = (-2*0) + (-1*3) = -3
Result Ma... |
Evaluate sin(90°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(90°) 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_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 = 8 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: 8^2 + 4^2 = c^2
2. Calculate the squares: 64 + 16 = c^2
3. Sum the squares: 80 = c^2
4. Take the square root of both sides: c = √80
Final Answer:... |
Add the fractions 8/3 and 6/8. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 3 and 8, which is 24.
2. Convert 8/3 by multiplying top and bottom by 8: 64/24.
3. Convert 6/8 by multiplying top and bottom by 3: 18/24.
4. Add the numerators: 64 + 18 = 82.
5. The resulting fraction is 82/24.
6. Simplify by div... |
Compute the quotient and remainder of 10513 divided by 82. Show the long division steps. | To divide 10513 by 82:
1. Determine how many times 82 goes into the leading digits of 10513.
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: 10513 / 82 = 128 with a remainder of 17.
Proof: (128 * 82) ... |
Find the roots of the quadratic equation: 6x^2 + -4x + 6 = 0 | 1. Identify coefficients: a=6, b=-4, c=6.
2. Calculate the discriminant: Δ = b^2 - 4ac = -4^2 - 4(6)(6) = 16 - 144 = -128.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (4 ± √128i) / 12.
Final Answer: x = 0.3333333333333333 ± 0.9428090415820635i |
In a right-angled triangle, the lengths of the two legs are a = 17 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: 17^2 + 12^2 = c^2
2. Calculate the squares: 289 + 144 = c^2
3. Sum the squares: 433 = c^2
4. Take the square root of both sides: c = √433
Final A... |
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 |
Compute the indefinite integral of f(x) = 2x^1. | To find the indefinite integral ∫ 2x^1 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 1 + 1 = 2.
2. Divide the coefficient by the new exponent: 2 / 2 = 1.
3. Combine and add the constant of integration C.
Final Answer: ∫ 2x^1 dx = 1x^2 + C |
In a right-angled triangle, the lengths of the two legs are a = 16 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: 16^2 + 10^2 = c^2
2. Calculate the squares: 256 + 100 = c^2
3. Sum the squares: 356 = c^2
4. Take the square root of both sides: c = √356
Final A... |
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: [19, 6, 12, 4, 4] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (19 + 6 + 12 + 4 + 4) / 5 = 45 / 5 = 9.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [10.0, -3.0, 3.0, -5.0, -5.0]
Squared Differences: [100.0, 9.0, 9.0, 25.0, 25.0]
Su... |
Multiply the following 2x2 matrices:
A =
[-4, 3]
[1, 2]
B =
[5, -3]
[0, -5] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-4*5) + (3*0) = -20
C12 = (A11 * B12) + (A12 * B22) = (-4*-3) + (3*-5) = -3
C21 = (A21 * B11) + (A22 * B21) = (1*5) + (2*0) = 5
C22 = (A21 * B12) + (A22 * B22) = (1*-3) + (2*-5) = -13
Result... |
Find the roots of the quadratic equation: 7x^2 + 0x + 6 = 0 | 1. Identify coefficients: a=7, b=0, c=6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 0^2 - 4(7)(6) = 0 - 168 = -168.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (0 ± √168i) / 14.
Final Answer: x = 0.0 ± 0.9258200997725515i |
Compute the quotient and remainder of 5796 divided by 51. Show the long division steps. | To divide 5796 by 51:
1. Determine how many times 51 goes into the leading digits of 5796.
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: 5796 / 51 = 113 with a remainder of 33.
Proof: (113 * 51) + 3... |
Calculate the mean, population variance, and standard deviation for the dataset: [8, 15, 6, 18, 15] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (8 + 15 + 6 + 18 + 15) / 5 = 62 / 5 = 12.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-4.4, 2.6, -6.4, 5.6, 2.6]
Squared Differences: [19.36, 6.76, 40.96, 31.36, 6.76]
... |
Find the roots of the quadratic equation: 10x^2 + 5x + 1 = 0 | 1. Identify coefficients: a=10, b=5, c=1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 5^2 - 4(10)(1) = 25 - 40 = -15.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-5 ± √15i) / 20.
Final Answer: x = -0.25 ± 0.19364916731037085i |
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 +... |
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 |
Find the derivative of f(x) = (4x + 4)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 4x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 4. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 4)^2 * 4 = 12(4x ... |
Add the fractions 4/3 and 3/5. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 3 and 5, which is 15.
2. Convert 4/3 by multiplying top and bottom by 5: 20/15.
3. Convert 3/5 by multiplying top and bottom by 3: 9/15.
4. Add the numerators: 20 + 9 = 29.
5. The resulting fraction is 29/15.
Final Answer: 29/15 |
Compute the quotient and remainder of 4329 divided by 19. Show the long division steps. | To divide 4329 by 19:
1. Determine how many times 19 goes into the leading digits of 4329.
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: 4329 / 19 = 227 with a remainder of 16.
Proof: (227 * 19) + 1... |
Find the roots of the quadratic equation: 8x^2 + -5x + -6 = 0 | 1. Identify coefficients: a=8, b=-5, c=-6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 25 - -192 = 217.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (5 ± √217) / 16.
5. Calculate roots: x1 = 1.2331824914160148, x2 = -0.6081824914160147.
Final... |
Evaluate sin(45°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(45°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.7071 (exact form depends on root simplification if applicable) |
Compute the quotient and remainder of 87293 divided by 79. Show the long division steps. | To divide 87293 by 79:
1. Determine how many times 79 goes into the leading digits of 87293.
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: 87293 / 79 = 1104 with a remainder of 77.
Proof: (1104 * 79... |
Find the roots of the quadratic equation: 9x^2 + -8x + 8 = 0 | 1. Identify coefficients: a=9, b=-8, c=8.
2. Calculate the discriminant: Δ = b^2 - 4ac = -8^2 - 4(9)(8) = 64 - 288 = -224.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (8 ± √224i) / 18.
Final Answer: x = 0.4444444444444444 ± 0.8314794192830981i |
Find the roots of the quadratic equation: 10x^2 + 2x + -7 = 0 | 1. Identify coefficients: a=10, b=2, c=-7.
2. Calculate the discriminant: Δ = b^2 - 4ac = 4 - -280 = 284.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-2 ± √284) / 20.
5. Calculate roots: x1 = 0.7426149773176359, x2 = -0.9426149773176359.
Final... |
Multiply the following 2x2 matrices:
A =
[-5, -2]
[4, 2]
B =
[5, -4]
[3, 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) = (-5*5) + (-2*3) = -31
C12 = (A11 * B12) + (A12 * B22) = (-5*-4) + (-2*4) = 12
C21 = (A21 * B11) + (A22 * B21) = (4*5) + (2*3) = 26
C22 = (A21 * B12) + (A22 * B22) = (4*-4) + (2*4) = -8
Result... |
Compute the indefinite integral of f(x) = 2x^2. | To find the indefinite integral ∫ 2x^2 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 2 + 1 = 3.
2. Divide the coefficient by the new exponent: 2 / 3 = 2/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 2x^2 dx = 2/3x^3 + C |
Compute the indefinite integral of f(x) = 8x^3. | To find the indefinite integral ∫ 8x^3 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 3 + 1 = 4.
2. Divide the coefficient by the new exponent: 8 / 4 = 2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 8x^3 dx = 2x^4 + C |
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 ... |
Find the roots of the quadratic equation: 9x^2 + 6x + -8 = 0 | 1. Identify coefficients: a=9, b=6, c=-8.
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) / 18.
5. Calculate roots: x1 = 0.6666666666666666, x2 = -1.3333333333333333.
Final... |
Find the roots of the quadratic equation: 8x^2 + -3x + -2 = 0 | 1. Identify coefficients: a=8, b=-3, c=-2.
2. Calculate the discriminant: Δ = b^2 - 4ac = 9 - -64 = 73.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (3 ± √73) / 16.
5. Calculate roots: x1 = 0.7215002340823456, x2 = -0.34650023408234565.
Final An... |
Compute the quotient and remainder of 51777 divided by 8. Show the long division steps. | To divide 51777 by 8:
1. Determine how many times 8 goes into the leading digits of 51777.
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: 51777 / 8 = 6472 with a remainder of 1.
Proof: (6472 * 8) + 1... |
Find the roots of the quadratic equation: 8x^2 + -7x + 7 = 0 | 1. Identify coefficients: a=8, b=-7, c=7.
2. Calculate the discriminant: Δ = b^2 - 4ac = -7^2 - 4(8)(7) = 49 - 224 = -175.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (7 ± √175i) / 16.
Final Answer: x = 0.4375 ± 0.8267972847076845i |
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) |
Add the fractions 15/2 and 13/14. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 2 and 14, which is 14.
2. Convert 15/2 by multiplying top and bottom by 7: 105/14.
3. Convert 13/14 by multiplying top and bottom by 1: 13/14.
4. Add the numerators: 105 + 13 = 118.
5. The resulting fraction is 118/14.
6. Simplif... |
Find the derivative of f(x) = (4x + 2)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 4x + 2.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 2. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 2)^2 * 4 = 12(4x ... |
Evaluate the logarithm: log_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 |
Find the roots of the quadratic equation: 10x^2 + 6x + -10 = 0 | 1. Identify coefficients: a=10, b=6, c=-10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 36 - -400 = 436.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-6 ± √436) / 20.
5. Calculate roots: x1 = 0.744030650891055, x2 = -1.3440306508910551.
Fina... |
Compute the quotient and remainder of 12161 divided by 95. Show the long division steps. | To divide 12161 by 95:
1. Determine how many times 95 goes into the leading digits of 12161.
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: 12161 / 95 = 128 with a remainder of 1.
Proof: (128 * 95) +... |
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