prompt stringlengths 32 115 | completion stringlengths 167 566 |
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Multiply the following 2x2 matrices:
A =
[5, 2]
[-4, 5]
B =
[-1, -4]
[-5, -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*-1) + (2*-5) = -15
C12 = (A11 * B12) + (A12 * B22) = (5*-4) + (2*-4) = -28
C21 = (A21 * B11) + (A22 * B21) = (-4*-1) + (5*-5) = -21
C22 = (A21 * B12) + (A22 * B22) = (-4*-4) + (5*-4) = -4
... |
Find the derivative of f(x) = (5x + 4)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 5x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 5x + 4. Its derivative is h'(x) = 5.
4. Multiply g'(u) by h'(x): f'(x) = 3(5x + 4)^2 * 5 = 15(5x ... |
Find the 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) = 8x^1. | To find the indefinite integral ∫ 8x^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: 8 / 2 = 4.
3. Combine and add the constant of integration C.
Final Answer: ∫ 8x^1 dx = 4x^2 + C |
Find the roots of the quadratic equation: 5x^2 + -2x + 8 = 0 | 1. Identify coefficients: a=5, b=-2, c=8.
2. Calculate the discriminant: Δ = b^2 - 4ac = -2^2 - 4(5)(8) = 4 - 160 = -156.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (2 ± √156i) / 10.
Final Answer: x = 0.2 ± 1.2489995996796797i |
Add the fractions 4/3 and 9/4. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 3 and 4, which is 12.
2. Convert 4/3 by multiplying top and bottom by 4: 16/12.
3. Convert 9/4 by multiplying top and bottom by 3: 27/12.
4. Add the numerators: 16 + 27 = 43.
5. The resulting fraction is 43/12.
Final Answer: 43/1... |
Multiply the following 2x2 matrices:
A =
[1, -2]
[4, 3]
B =
[5, 2]
[2, 0] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (1*5) + (-2*2) = 1
C12 = (A11 * B12) + (A12 * B22) = (1*2) + (-2*0) = 2
C21 = (A21 * B11) + (A22 * B21) = (4*5) + (3*2) = 26
C22 = (A21 * B12) + (A22 * B22) = (4*2) + (3*0) = 8
Result Matrix ... |
Compute the quotient and remainder of 24306 divided by 58. Show the long division steps. | To divide 24306 by 58:
1. Determine how many times 58 goes into the leading digits of 24306.
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: 24306 / 58 = 419 with a remainder of 4.
Proof: (419 * 58) +... |
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 |
Compute the indefinite integral of f(x) = 3x^5. | To find the indefinite integral ∫ 3x^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: 3 / 6 = 1/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 3x^5 dx = 1/2x^6 + C |
Add the fractions 10/11 and 2/15. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 11 and 15, which is 165.
2. Convert 10/11 by multiplying top and bottom by 15: 150/165.
3. Convert 2/15 by multiplying top and bottom by 11: 22/165.
4. Add the numerators: 150 + 22 = 172.
5. The resulting fraction is 172/165.
Fin... |
Add the fractions 3/5 and 12/10. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 5 and 10, which is 10.
2. Convert 3/5 by multiplying top and bottom by 2: 6/10.
3. Convert 12/10 by multiplying top and bottom by 1: 12/10.
4. Add the numerators: 6 + 12 = 18.
5. The resulting fraction is 18/10.
6. Simplify by di... |
Calculate the mean, population variance, and standard deviation for the dataset: [6, 12, 9, 18, 20] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (6 + 12 + 9 + 18 + 20) / 5 = 65 / 5 = 13.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-7.0, -1.0, -4.0, 5.0, 7.0]
Squared Differences: [49.0, 1.0, 16.0, 25.0, 49.0]
S... |
Compute the quotient and remainder of 50994 divided by 38. Show the long division steps. | To divide 50994 by 38:
1. Determine how many times 38 goes into the leading digits of 50994.
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: 50994 / 38 = 1341 with a remainder of 36.
Proof: (1341 * 38... |
Multiply the following 2x2 matrices:
A =
[3, -5]
[5, 5]
B =
[1, 2]
[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) = (3*1) + (-5*3) = -12
C12 = (A11 * B12) + (A12 * B22) = (3*2) + (-5*-4) = 26
C21 = (A21 * B11) + (A22 * B21) = (5*1) + (5*3) = 20
C22 = (A21 * B12) + (A22 * B22) = (5*2) + (5*-4) = -10
Result ... |
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 |
Multiply the following 2x2 matrices:
A =
[1, 4]
[-5, 0]
B =
[3, 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*3) + (4*2) = 11
C12 = (A11 * B12) + (A12 * B22) = (1*5) + (4*-1) = 1
C21 = (A21 * B11) + (A22 * B21) = (-5*3) + (0*2) = -15
C22 = (A21 * B12) + (A22 * B22) = (-5*5) + (0*-1) = -25
Result M... |
Compute the indefinite integral of f(x) = 3x^2. | To find the indefinite integral ∫ 3x^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: 3 / 3 = 1.
3. Combine and add the constant of integration C.
Final Answer: ∫ 3x^2 dx = 1x^3 + C |
Add the fractions 15/8 and 2/2. 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 2/2 by multiplying top and bottom by 4: 8/8.
4. Add the numerators: 15 + 8 = 23.
5. The resulting fraction is 23/8.
Final Answer: 23/8 |
Find the roots of the quadratic equation: 9x^2 + 7x + -9 = 0 | 1. Identify coefficients: a=9, b=7, c=-9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 49 - -324 = 373.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-7 ± √373) / 18.
5. Calculate roots: x1 = 0.684067106434887, x2 = -1.4618448842126648.
Final ... |
In a right-angled triangle, the lengths of the two legs are a = 10 and b = 7. 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 + 7^2 = c^2
2. Calculate the squares: 100 + 49 = c^2
3. Sum the squares: 149 = c^2
4. Take the square root of both sides: c = √149
Final Ans... |
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 |
Add the fractions 11/15 and 3/7. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 15 and 7, which is 105.
2. Convert 11/15 by multiplying top and bottom by 7: 77/105.
3. Convert 3/7 by multiplying top and bottom by 15: 45/105.
4. Add the numerators: 77 + 45 = 122.
5. The resulting fraction is 122/105.
Final An... |
Find the roots of the quadratic equation: 7x^2 + -3x + 3 = 0 | 1. Identify coefficients: a=7, b=-3, c=3.
2. Calculate the discriminant: Δ = b^2 - 4ac = -3^2 - 4(7)(3) = 9 - 84 = -75.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (3 ± √75i) / 14.
Final Answer: x = 0.21428571428571427 ± 0.618589574131742i |
Add the fractions 6/13 and 2/12. 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 6/13 by multiplying top and bottom by 12: 72/156.
3. Convert 2/12 by multiplying top and bottom by 13: 26/156.
4. Add the numerators: 72 + 26 = 98.
5. The resulting fraction is 98/156.
6. Simpl... |
Multiply the following 2x2 matrices:
A =
[3, -3]
[2, -3]
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) = (3*5) + (-3*-1) = 18
C12 = (A11 * B12) + (A12 * B22) = (3*5) + (-3*-3) = 24
C21 = (A21 * B11) + (A22 * B21) = (2*5) + (-3*-1) = 13
C22 = (A21 * B12) + (A22 * B22) = (2*5) + (-3*-3) = 19
Resul... |
Find the derivative of f(x) = (3x + 5)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 3x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 5. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 5)^2 * 3 = 9(3x +... |
Calculate the mean, population variance, and standard deviation for the dataset: [4, 11, 3, 5, 2] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (4 + 11 + 3 + 5 + 2) / 5 = 25 / 5 = 5.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-1.0, 6.0, -2.0, 0.0, -3.0]
Squared Differences: [1.0, 36.0, 4.0, 0.0, 9.0]
Sum of ... |
Compute the quotient and remainder of 33942 divided by 14. Show the long division steps. | To divide 33942 by 14:
1. Determine how many times 14 goes into the leading digits of 33942.
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: 33942 / 14 = 2424 with a remainder of 6.
Proof: (2424 * 14)... |
Multiply the following 2x2 matrices:
A =
[2, -3]
[-1, -4]
B =
[3, -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) = (2*3) + (-3*2) = 0
C12 = (A11 * B12) + (A12 * B22) = (2*-5) + (-3*1) = -13
C21 = (A21 * B11) + (A22 * B21) = (-1*3) + (-4*2) = -11
C22 = (A21 * B12) + (A22 * B22) = (-1*-5) + (-4*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 +... |
Calculate the mean, population variance, and standard deviation for the dataset: [18, 1, 10, 4, 15] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (18 + 1 + 10 + 4 + 15) / 5 = 48 / 5 = 9.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [8.4, -8.6, 0.4, -5.6, 5.4]
Squared Differences: [70.56, 73.96, 0.16, 31.36, 29.16]
... |
In a right-angled triangle, the lengths of the two legs are a = 13 and b = 9. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 13^2 + 9^2 = c^2
2. Calculate the squares: 169 + 81 = c^2
3. Sum the squares: 250 = c^2
4. Take the square root of both sides: c = √250
Final Ans... |
Multiply the following 2x2 matrices:
A =
[3, 4]
[-2, -4]
B =
[4, -4]
[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) = (3*4) + (4*2) = 20
C12 = (A11 * B12) + (A12 * B22) = (3*-4) + (4*3) = 0
C21 = (A21 * B11) + (A22 * B21) = (-2*4) + (-4*2) = -16
C22 = (A21 * B12) + (A22 * B22) = (-2*-4) + (-4*3) = -4
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 ... |
Find the roots of the quadratic equation: 8x^2 + -7x + -1 = 0 | 1. Identify coefficients: a=8, b=-7, c=-1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 49 - -32 = 81.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (7 ± √81) / 16.
5. Calculate roots: x1 = 1.0, x2 = -0.125.
Final Answer: x = 1.0, -0.125 |
Calculate the mean, population variance, and standard deviation for the dataset: [9, 9, 13, 17, 14] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (9 + 9 + 13 + 17 + 14) / 5 = 62 / 5 = 12.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-3.4, -3.4, 0.6, 4.6, 1.6]
Squared Differences: [11.56, 11.56, 0.36, 21.16, 2.56]
... |
Add the fractions 5/3 and 7/10. 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 5/3 by multiplying top and bottom by 10: 50/30.
3. Convert 7/10 by multiplying top and bottom by 3: 21/30.
4. Add the numerators: 50 + 21 = 71.
5. The resulting fraction is 71/30.
Final Answer: 7... |
Evaluate sin(60°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(60°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.8660 (exact form depends on root simplification if applicable) |
Compute the quotient and remainder of 80885 divided by 37. Show the long division steps. | To divide 80885 by 37:
1. Determine how many times 37 goes into the leading digits of 80885.
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: 80885 / 37 = 2186 with a remainder of 3.
Proof: (2186 * 37)... |
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) |
Calculate the mean, population variance, and standard deviation for the dataset: [19, 17, 12, 20, 11] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (19 + 17 + 12 + 20 + 11) / 5 = 79 / 5 = 15.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [3.2, 1.2, -3.8, 4.2, -4.8]
Squared Differences: [10.24, 1.44, 14.44, 17.64, 23.04... |
Evaluate sin(0°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(0°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.0000 (exact form depends on root simplification if applicable) |
Compute the quotient and remainder of 63510 divided by 67. Show the long division steps. | To divide 63510 by 67:
1. Determine how many times 67 goes into the leading digits of 63510.
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: 63510 / 67 = 947 with a remainder of 61.
Proof: (947 * 67) ... |
Multiply the following 2x2 matrices:
A =
[5, 1]
[-2, -4]
B =
[-4, 1]
[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) = (5*-4) + (1*0) = -20
C12 = (A11 * B12) + (A12 * B22) = (5*1) + (1*5) = 10
C21 = (A21 * B11) + (A22 * B21) = (-2*-4) + (-4*0) = 8
C22 = (A21 * B12) + (A22 * B22) = (-2*1) + (-4*5) = -22
Result... |
Multiply the following 2x2 matrices:
A =
[-3, 1]
[-2, 4]
B =
[-3, -2]
[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) = (-3*-3) + (1*2) = 11
C12 = (A11 * B12) + (A12 * B22) = (-3*-2) + (1*-3) = 3
C21 = (A21 * B11) + (A22 * B21) = (-2*-3) + (4*2) = 14
C22 = (A21 * B12) + (A22 * B22) = (-2*-2) + (4*-3) = -8
Resu... |
Compute the indefinite integral of f(x) = 2x^4. | To find the indefinite integral ∫ 2x^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: 2 / 5 = 2/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 2x^4 dx = 2/5x^5 + C |
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 |
Add the fractions 8/8 and 4/15. 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 8/8 by multiplying top and bottom by 15: 120/120.
3. Convert 4/15 by multiplying top and bottom by 8: 32/120.
4. Add the numerators: 120 + 32 = 152.
5. The resulting fraction is 152/120.
6. Simp... |
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 ... |
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 +... |
Calculate the mean, population variance, and standard deviation for the dataset: [6, 11, 9, 5, 11] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (6 + 11 + 9 + 5 + 11) / 5 = 42 / 5 = 8.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-2.4, 2.6, 0.6, -3.4, 2.6]
Squared Differences: [5.76, 6.76, 0.36, 11.56, 6.76]
Su... |
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 |
Calculate the mean, population variance, and standard deviation for the dataset: [18, 3, 20, 8, 15] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (18 + 3 + 20 + 8 + 15) / 5 = 64 / 5 = 12.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [5.2, -9.8, 7.2, -4.8, 2.2]
Squared Differences: [27.04, 96.04, 51.84, 23.04, 4.84]
... |
Find the roots of the quadratic equation: 2x^2 + -7x + 4 = 0 | 1. Identify coefficients: a=2, b=-7, c=4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 49 - 32 = 17.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (7 ± √17) / 4.
5. Calculate roots: x1 = 2.7807764064044154, x2 = 0.7192235935955849.
Final Answer... |
Multiply the following 2x2 matrices:
A =
[5, -1]
[-4, 0]
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) = (5*1) + (-1*-2) = 7
C12 = (A11 * B12) + (A12 * B22) = (5*3) + (-1*-3) = 18
C21 = (A21 * B11) + (A22 * B21) = (-4*1) + (0*-2) = -4
C22 = (A21 * B12) + (A22 * B22) = (-4*3) + (0*-3) = -12
Resul... |
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 |
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) |
Multiply the following 2x2 matrices:
A =
[2, 5]
[-1, -1]
B =
[1, 1]
[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) = (2*1) + (5*1) = 7
C12 = (A11 * B12) + (A12 * B22) = (2*1) + (5*-2) = -8
C21 = (A21 * B11) + (A22 * B21) = (-1*1) + (-1*1) = -2
C22 = (A21 * B12) + (A22 * B22) = (-1*1) + (-1*-2) = 1
Result Ma... |
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 |
Compute the quotient and remainder of 88750 divided by 26. Show the long division steps. | To divide 88750 by 26:
1. Determine how many times 26 goes into the leading digits of 88750.
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: 88750 / 26 = 3413 with a remainder of 12.
Proof: (3413 * 26... |
Add the fractions 10/12 and 10/3. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 12 and 3, which is 12.
2. Convert 10/12 by multiplying top and bottom by 1: 10/12.
3. Convert 10/3 by multiplying top and bottom by 4: 40/12.
4. Add the numerators: 10 + 40 = 50.
5. The resulting fraction is 50/12.
6. Simplify by... |
Evaluate the logarithm: log_5(3125) | To evaluate log_5(3125), we ask the question: '5 raised to what power equals 3125?'
Let x be the unknown power: 5^x = 3125
Since 5^5 = 3125, it follows that x = 5.
Final Answer: 5 |
Evaluate the logarithm: log_4(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 |
In a right-angled triangle, the lengths of the two legs are a = 7 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: 7^2 + 19^2 = c^2
2. Calculate the squares: 49 + 361 = c^2
3. Sum the squares: 410 = c^2
4. Take the square root of both sides: c = √410
Final Ans... |
Find the roots of the quadratic equation: 4x^2 + 10x + 5 = 0 | 1. Identify coefficients: a=4, b=10, c=5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 100 - 80 = 20.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-10 ± √20) / 8.
5. Calculate roots: x1 = -0.6909830056250525, x2 = -1.8090169943749475.
Final A... |
Multiply the following 2x2 matrices:
A =
[5, -5]
[2, 3]
B =
[-2, -2]
[2, 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) = (5*-2) + (-5*2) = -20
C12 = (A11 * B12) + (A12 * B22) = (5*-2) + (-5*0) = -10
C21 = (A21 * B11) + (A22 * B21) = (2*-2) + (3*2) = 2
C22 = (A21 * B12) + (A22 * B22) = (2*-2) + (3*0) = -4
Result... |
In a right-angled triangle, the lengths of the two legs are a = 18 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: 18^2 + 15^2 = c^2
2. Calculate the squares: 324 + 225 = c^2
3. Sum the squares: 549 = c^2
4. Take the square root of both sides: c = √549
Final A... |
Find the roots of the quadratic equation: 6x^2 + -3x + 8 = 0 | 1. Identify coefficients: a=6, b=-3, c=8.
2. Calculate the discriminant: Δ = b^2 - 4ac = -3^2 - 4(6)(8) = 9 - 192 = -183.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (3 ± √183i) / 12.
Final Answer: x = 0.25 ± 1.1273124382057236i |
Calculate the mean, population variance, and standard deviation for the dataset: [8, 9, 7, 13, 4] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (8 + 9 + 7 + 13 + 4) / 5 = 41 / 5 = 8.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-0.2, 0.8, -1.2, 4.8, -4.2]
Squared Differences: [0.04, 0.64, 1.44, 23.04, 17.64]
S... |
Evaluate the logarithm: log_5(625) | To evaluate log_5(625), we ask the question: '5 raised to what power equals 625?'
Let x be the unknown power: 5^x = 625
Since 5^4 = 625, it follows that x = 4.
Final Answer: 4 |
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) |
Compute the indefinite integral of f(x) = 9x^4. | To find the indefinite integral ∫ 9x^4 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 4 + 1 = 5.
2. Divide the coefficient by the new exponent: 9 / 5 = 9/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 9x^4 dx = 9/5x^5 + C |
Compute the indefinite integral of f(x) = 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 |
Compute the quotient and remainder of 55351 divided by 53. Show the long division steps. | To divide 55351 by 53:
1. Determine how many times 53 goes into the leading digits of 55351.
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: 55351 / 53 = 1044 with a remainder of 19.
Proof: (1044 * 53... |
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 ... |
Add the fractions 9/11 and 7/10. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 11 and 10, which is 110.
2. Convert 9/11 by multiplying top and bottom by 10: 90/110.
3. Convert 7/10 by multiplying top and bottom by 11: 77/110.
4. Add the numerators: 90 + 77 = 167.
5. The resulting fraction is 167/110.
Final ... |
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 ... |
In a right-angled triangle, the lengths of the two legs are a = 16 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: 16^2 + 17^2 = c^2
2. Calculate the squares: 256 + 289 = c^2
3. Sum the squares: 545 = c^2
4. Take the square root of both sides: c = √545
Final A... |
Multiply the following 2x2 matrices:
A =
[1, 1]
[1, -2]
B =
[-1, 1]
[0, 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) + (1*0) = -1
C12 = (A11 * B12) + (A12 * B22) = (1*1) + (1*1) = 2
C21 = (A21 * B11) + (A22 * B21) = (1*-1) + (-2*0) = -1
C22 = (A21 * B12) + (A22 * B22) = (1*1) + (-2*1) = -1
Result Mat... |
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) |
Multiply the following 2x2 matrices:
A =
[0, 1]
[2, 2]
B =
[-5, 2]
[0, 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) = (0*-5) + (1*0) = 0
C12 = (A11 * B12) + (A12 * B22) = (0*2) + (1*2) = 2
C21 = (A21 * B11) + (A22 * B21) = (2*-5) + (2*0) = -10
C22 = (A21 * B12) + (A22 * B22) = (2*2) + (2*2) = 8
Result Matrix... |
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 |
In a right-angled triangle, the lengths of the two legs are a = 14 and b = 7. 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 + 7^2 = c^2
2. Calculate the squares: 196 + 49 = c^2
3. Sum the squares: 245 = c^2
4. Take the square root of both sides: c = √245
Final Ans... |
Find the roots of the quadratic equation: 10x^2 + -1x + 8 = 0 | 1. Identify coefficients: a=10, b=-1, c=8.
2. Calculate the discriminant: Δ = b^2 - 4ac = -1^2 - 4(10)(8) = 1 - 320 = -319.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (1 ± √319i) / 20.
Final Answer: x = 0.05 ± 0.8930285549745876i |
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 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) |
Evaluate the logarithm: log_2(16) | To evaluate log_2(16), we ask the question: '2 raised to what power equals 16?'
Let x be the unknown power: 2^x = 16
Since 2^4 = 16, it follows that x = 4.
Final Answer: 4 |
Evaluate 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 quotient and remainder of 10667 divided by 89. Show the long division steps. | To divide 10667 by 89:
1. Determine how many times 89 goes into the leading digits of 10667.
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: 10667 / 89 = 119 with a remainder of 76.
Proof: (119 * 89) ... |
Find the roots of the quadratic equation: 10x^2 + -4x + -1 = 0 | 1. Identify coefficients: a=10, b=-4, c=-1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 16 - -40 = 56.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (4 ± √56) / 20.
5. Calculate roots: x1 = 0.5741657386773941, x2 = -0.17416573867739413.
Final ... |
Add the fractions 9/4 and 13/7. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 4 and 7, which is 28.
2. Convert 9/4 by multiplying top and bottom by 7: 63/28.
3. Convert 13/7 by multiplying top and bottom by 4: 52/28.
4. Add the numerators: 63 + 52 = 115.
5. The resulting fraction is 115/28.
Final Answer: 1... |
Add the fractions 2/11 and 7/4. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 11 and 4, which is 44.
2. Convert 2/11 by multiplying top and bottom by 4: 8/44.
3. Convert 7/4 by multiplying top and bottom by 11: 77/44.
4. Add the numerators: 8 + 77 = 85.
5. The resulting fraction is 85/44.
Final Answer: 85/... |
Find the roots of the quadratic equation: 1x^2 + -10x + -1 = 0 | 1. Identify coefficients: a=1, b=-10, c=-1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 100 - -4 = 104.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (10 ± √104) / 2.
5. Calculate roots: x1 = 10.099019513592784, x2 = -0.09901951359278449.
Fina... |
Compute the quotient and remainder of 40772 divided by 50. Show the long division steps. | To divide 40772 by 50:
1. Determine how many times 50 goes into the leading digits of 40772.
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: 40772 / 50 = 815 with a remainder of 22.
Proof: (815 * 50) ... |
In a right-angled triangle, the lengths of the two legs are a = 9 and b = 13. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 9^2 + 13^2 = c^2
2. Calculate the squares: 81 + 169 = c^2
3. Sum the squares: 250 = c^2
4. Take the square root of both sides: c = √250
Final Ans... |
In a right-angled triangle, the lengths of the two legs are a = 19 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: 19^2 + 3^2 = c^2
2. Calculate the squares: 361 + 9 = c^2
3. Sum the squares: 370 = c^2
4. Take the square root of both sides: c = √370
Final Answ... |
Add the fractions 2/8 and 9/5. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 8 and 5, which is 40.
2. Convert 2/8 by multiplying top and bottom by 5: 10/40.
3. Convert 9/5 by multiplying top and bottom by 8: 72/40.
4. Add the numerators: 10 + 72 = 82.
5. The resulting fraction is 82/40.
6. Simplify by div... |
Compute the quotient and remainder of 59800 divided by 7. Show the long division steps. | To divide 59800 by 7:
1. Determine how many times 7 goes into the leading digits of 59800.
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: 59800 / 7 = 8542 with a remainder of 6.
Proof: (8542 * 7) + 6... |
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 |
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