prompt stringlengths 32 115 | completion stringlengths 167 566 |
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Multiply the following 2x2 matrices:
A =
[1, 5]
[3, 0]
B =
[1, 4]
[2, 1] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (1*1) + (5*2) = 11
C12 = (A11 * B12) + (A12 * B22) = (1*4) + (5*1) = 9
C21 = (A21 * B11) + (A22 * B21) = (3*1) + (0*2) = 3
C22 = (A21 * B12) + (A22 * B22) = (3*4) + (0*1) = 12
Result Matrix C... |
Compute the quotient and remainder of 19371 divided by 36. Show the long division steps. | To divide 19371 by 36:
1. Determine how many times 36 goes into the leading digits of 19371.
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: 19371 / 36 = 538 with a remainder of 3.
Proof: (538 * 36) +... |
Calculate the mean, population variance, and standard deviation for the dataset: [15, 20, 7, 17, 9] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (15 + 20 + 7 + 17 + 9) / 5 = 68 / 5 = 13.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [1.4, 6.4, -6.6, 3.4, -4.6]
Squared Differences: [1.96, 40.96, 43.56, 11.56, 21.16]
... |
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... |
Compute the quotient and remainder of 18053 divided by 5. Show the long division steps. | To divide 18053 by 5:
1. Determine how many times 5 goes into the leading digits of 18053.
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: 18053 / 5 = 3610 with a remainder of 3.
Proof: (3610 * 5) + 3... |
Add the fractions 5/8 and 4/11. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 8 and 11, which is 88.
2. Convert 5/8 by multiplying top and bottom by 11: 55/88.
3. Convert 4/11 by multiplying top and bottom by 8: 32/88.
4. Add the numerators: 55 + 32 = 87.
5. The resulting fraction is 87/88.
Final Answer: 8... |
Evaluate the logarithm: log_4(16) | To evaluate log_4(16), we ask the question: '4 raised to what power equals 16?'
Let x be the unknown power: 4^x = 16
Since 4^2 = 16, it follows that x = 2.
Final Answer: 2 |
Compute the indefinite integral of f(x) = 1x^1. | To find the indefinite integral ∫ 1x^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: 1 / 2 = 1/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 1x^1 dx = 1/2x^2 + C |
Multiply the following 2x2 matrices:
A =
[4, -4]
[-3, -2]
B =
[1, -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) = (4*1) + (-4*4) = -12
C12 = (A11 * B12) + (A12 * B22) = (4*-5) + (-4*-5) = 0
C21 = (A21 * B11) + (A22 * B21) = (-3*1) + (-2*4) = -11
C22 = (A21 * B12) + (A22 * B22) = (-3*-5) + (-2*-5) = 25
Re... |
In a right-angled triangle, the lengths of the two legs are a = 18 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: 18^2 + 19^2 = c^2
2. Calculate the squares: 324 + 361 = c^2
3. Sum the squares: 685 = c^2
4. Take the square root of both sides: c = √685
Final A... |
Calculate the mean, population variance, and standard deviation for the dataset: [10, 14, 17, 1, 15] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (10 + 14 + 17 + 1 + 15) / 5 = 57 / 5 = 11.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-1.4, 2.6, 5.6, -10.4, 3.6]
Squared Differences: [1.96, 6.76, 31.36, 108.16, 12.96... |
Find the roots of the quadratic equation: 10x^2 + 10x + 4 = 0 | 1. Identify coefficients: a=10, b=10, c=4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 10^2 - 4(10)(4) = 100 - 160 = -60.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-10 ± √60i) / 20.
Final Answer: x = -0.5 ± 0.3872983346207417i |
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... |
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 |
Multiply the following 2x2 matrices:
A =
[3, -1]
[0, 4]
B =
[3, -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*3) + (-1*-3) = 12
C12 = (A11 * B12) + (A12 * B22) = (3*-2) + (-1*4) = -10
C21 = (A21 * B11) + (A22 * B21) = (0*3) + (4*-3) = -12
C22 = (A21 * B12) + (A22 * B22) = (0*-2) + (4*4) = 16
Resul... |
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) |
Add the fractions 12/2 and 3/4. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 2 and 4, which is 4.
2. Convert 12/2 by multiplying top and bottom by 2: 24/4.
3. Convert 3/4 by multiplying top and bottom by 1: 3/4.
4. Add the numerators: 24 + 3 = 27.
5. The resulting fraction is 27/4.
Final Answer: 27/4 |
Calculate the mean, population variance, and standard deviation for the dataset: [9, 10, 20, 5, 3] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (9 + 10 + 20 + 5 + 3) / 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, 0.6, 10.6, -4.4, -6.4]
Squared Differences: [0.16, 0.36, 112.36, 19.36, 40.96]
... |
Calculate the mean, population variance, and standard deviation for the dataset: [20, 11, 6, 16, 3] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (20 + 11 + 6 + 16 + 3) / 5 = 56 / 5 = 11.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [8.8, -0.2, -5.2, 4.8, -8.2]
Squared Differences: [77.44, 0.04, 27.04, 23.04, 67.24]... |
Compute the indefinite integral of f(x) = 8x^2. | To find the indefinite integral ∫ 8x^2 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 2 + 1 = 3.
2. Divide the coefficient by the new exponent: 8 / 3 = 8/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 8x^2 dx = 8/3x^3 + C |
Compute the indefinite integral of f(x) = 9x^1. | To find the indefinite integral ∫ 9x^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: 9 / 2 = 9/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 9x^1 dx = 9/2x^2 + C |
Add the fractions 8/13 and 1/13. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 13 and 13, which is 13.
2. Convert 8/13 by multiplying top and bottom by 1: 8/13.
3. Convert 1/13 by multiplying top and bottom by 1: 1/13.
4. Add the numerators: 8 + 1 = 9.
5. The resulting fraction is 9/13.
Final Answer: 9/13 |
Add the fractions 5/14 and 15/4. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 14 and 4, which is 28.
2. Convert 5/14 by multiplying top and bottom by 2: 10/28.
3. Convert 15/4 by multiplying top and bottom by 7: 105/28.
4. Add the numerators: 10 + 105 = 115.
5. The resulting fraction is 115/28.
Final Answe... |
Multiply the following 2x2 matrices:
A =
[3, 5]
[5, 0]
B =
[3, -5]
[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) = (3*3) + (5*0) = 9
C12 = (A11 * B12) + (A12 * B22) = (3*-5) + (5*-2) = -25
C21 = (A21 * B11) + (A22 * B21) = (5*3) + (0*0) = 15
C22 = (A21 * B12) + (A22 * B22) = (5*-5) + (0*-2) = -25
Result M... |
Find the roots of the quadratic equation: 9x^2 + -1x + -6 = 0 | 1. Identify coefficients: a=9, b=-1, c=-6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1 - -216 = 217.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (1 ± √217) / 18.
5. Calculate roots: x1 = 0.8739399923697908, x2 = -0.7628288812586796.
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 ... |
Compute the quotient and remainder of 96923 divided by 89. Show the long division steps. | To divide 96923 by 89:
1. Determine how many times 89 goes into the leading digits of 96923.
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: 96923 / 89 = 1089 with a remainder of 2.
Proof: (1089 * 89)... |
Compute the quotient and remainder of 16909 divided by 85. Show the long division steps. | To divide 16909 by 85:
1. Determine how many times 85 goes into the leading digits of 16909.
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: 16909 / 85 = 198 with a remainder of 79.
Proof: (198 * 85) ... |
Find the roots of the quadratic equation: 2x^2 + -4x + -3 = 0 | 1. Identify coefficients: a=2, b=-4, c=-3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 16 - -24 = 40.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (4 ± √40) / 4.
5. Calculate roots: x1 = 2.58113883008419, x2 = -0.5811388300841898.
Final Answe... |
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 quotient and remainder of 27913 divided by 81. Show the long division steps. | To divide 27913 by 81:
1. Determine how many times 81 goes into the leading digits of 27913.
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: 27913 / 81 = 344 with a remainder of 49.
Proof: (344 * 81) ... |
Find the roots of the quadratic equation: 4x^2 + 8x + -4 = 0 | 1. Identify coefficients: a=4, b=8, c=-4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 64 - -64 = 128.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-8 ± √128) / 8.
5. Calculate roots: x1 = 0.41421356237309515, x2 = -2.414213562373095.
Final A... |
In a right-angled triangle, the lengths of the two legs are a = 4 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: 4^2 + 11^2 = c^2
2. Calculate the squares: 16 + 121 = c^2
3. Sum the squares: 137 = c^2
4. Take the square root of both sides: c = √137
Final Ans... |
Multiply the following 2x2 matrices:
A =
[0, 0]
[-1, 0]
B =
[-3, 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) = (0*-3) + (0*-3) = 0
C12 = (A11 * B12) + (A12 * B22) = (0*4) + (0*4) = 0
C21 = (A21 * B11) + (A22 * B21) = (-1*-3) + (0*-3) = 3
C22 = (A21 * B12) + (A22 * B22) = (-1*4) + (0*4) = -4
Result Mat... |
Add the fractions 10/9 and 6/8. 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 10/9 by multiplying top and bottom by 8: 80/72.
3. Convert 6/8 by multiplying top and bottom by 9: 54/72.
4. Add the numerators: 80 + 54 = 134.
5. The resulting fraction is 134/72.
6. Simplify by ... |
Compute the quotient and remainder of 59336 divided by 59. Show the long division steps. | To divide 59336 by 59:
1. Determine how many times 59 goes into the leading digits of 59336.
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: 59336 / 59 = 1005 with a remainder of 41.
Proof: (1005 * 59... |
Multiply the following 2x2 matrices:
A =
[-5, 4]
[-5, 5]
B =
[-1, 4]
[-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) = (-5*-1) + (4*-2) = -3
C12 = (A11 * B12) + (A12 * B22) = (-5*4) + (4*5) = 0
C21 = (A21 * B11) + (A22 * B21) = (-5*-1) + (5*-2) = -5
C22 = (A21 * B12) + (A22 * B22) = (-5*4) + (5*5) = 5
Result ... |
Compute the quotient and remainder of 93839 divided by 95. Show the long division steps. | To divide 93839 by 95:
1. Determine how many times 95 goes into the leading digits of 93839.
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: 93839 / 95 = 987 with a remainder of 74.
Proof: (987 * 95) ... |
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) |
Calculate the mean, population variance, and standard deviation for the dataset: [11, 18, 17, 2, 5] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (11 + 18 + 17 + 2 + 5) / 5 = 53 / 5 = 10.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [0.4, 7.4, 6.4, -8.6, -5.6]
Squared Differences: [0.16, 54.76, 40.96, 73.96, 31.36]
... |
Compute the indefinite integral of f(x) = 7x^1. | To find the indefinite integral ∫ 7x^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: 7 / 2 = 7/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 7x^1 dx = 7/2x^2 + C |
Find the roots of the quadratic equation: 7x^2 + 0x + -2 = 0 | 1. Identify coefficients: a=7, b=0, c=-2.
2. Calculate the discriminant: Δ = b^2 - 4ac = 0 - -56 = 56.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (0 ± √56) / 14.
5. Calculate roots: x1 = 0.5345224838248488, x2 = -0.5345224838248488.
Final Answ... |
Add the fractions 6/11 and 3/6. 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 6/11 by multiplying top and bottom by 6: 36/66.
3. Convert 3/6 by multiplying top and bottom by 11: 33/66.
4. Add the numerators: 36 + 33 = 69.
5. The resulting fraction is 69/66.
6. Simplify by ... |
Compute the quotient and remainder of 78821 divided by 25. Show the long division steps. | To divide 78821 by 25:
1. Determine how many times 25 goes into the leading digits of 78821.
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: 78821 / 25 = 3152 with a remainder of 21.
Proof: (3152 * 25... |
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: [15, 14, 5, 11, 11] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (15 + 14 + 5 + 11 + 11) / 5 = 56 / 5 = 11.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [3.8, 2.8, -6.2, -0.2, -0.2]
Squared Differences: [14.44, 7.84, 38.44, 0.04, 0.04]
... |
Compute the quotient and remainder of 38738 divided by 27. Show the long division steps. | To divide 38738 by 27:
1. Determine how many times 27 goes into the leading digits of 38738.
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: 38738 / 27 = 1434 with a remainder of 20.
Proof: (1434 * 27... |
Multiply the following 2x2 matrices:
A =
[2, 4]
[-1, -3]
B =
[-5, 5]
[-4, 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) = (2*-5) + (4*-4) = -26
C12 = (A11 * B12) + (A12 * B22) = (2*5) + (4*0) = 10
C21 = (A21 * B11) + (A22 * B21) = (-1*-5) + (-3*-4) = 17
C22 = (A21 * B12) + (A22 * B22) = (-1*5) + (-3*0) = -5
Resu... |
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... |
Multiply the following 2x2 matrices:
A =
[0, -4]
[0, -5]
B =
[3, -4]
[-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) = (0*3) + (-4*-2) = 8
C12 = (A11 * B12) + (A12 * B22) = (0*-4) + (-4*5) = -20
C21 = (A21 * B11) + (A22 * B21) = (0*3) + (-5*-2) = 10
C22 = (A21 * B12) + (A22 * B22) = (0*-4) + (-5*5) = -25
Resu... |
Compute the indefinite integral of f(x) = 5x^5. | To find the indefinite integral ∫ 5x^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: 5 / 6 = 5/6.
3. Combine and add the constant of integration C.
Final Answer: ∫ 5x^5 dx = 5/6x^6 + C |
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 |
Calculate the mean, population variance, and standard deviation for the dataset: [12, 12, 19, 17, 11] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (12 + 12 + 19 + 17 + 11) / 5 = 71 / 5 = 14.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-2.2, -2.2, 4.8, 2.8, -3.2]
Squared Differences: [4.84, 4.84, 23.04, 7.84, 10.24]... |
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 +... |
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 |
Multiply the following 2x2 matrices:
A =
[2, 3]
[4, -1]
B =
[-4, -5]
[-2, -2] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (2*-4) + (3*-2) = -14
C12 = (A11 * B12) + (A12 * B22) = (2*-5) + (3*-2) = -16
C21 = (A21 * B11) + (A22 * B21) = (4*-4) + (-1*-2) = -14
C22 = (A21 * B12) + (A22 * B22) = (4*-5) + (-1*-2) = -18
... |
Add the fractions 13/11 and 7/2. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 11 and 2, which is 22.
2. Convert 13/11 by multiplying top and bottom by 2: 26/22.
3. Convert 7/2 by multiplying top and bottom by 11: 77/22.
4. Add the numerators: 26 + 77 = 103.
5. The resulting fraction is 103/22.
Final Answer... |
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 |
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 indefinite integral of f(x) = 1x^3. | To find the indefinite integral ∫ 1x^3 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 3 + 1 = 4.
2. Divide the coefficient by the new exponent: 1 / 4 = 1/4.
3. Combine and add the constant of integration C.
Final Answer: ∫ 1x^3 dx = 1/4x^4 + C |
Find the roots of the quadratic equation: 2x^2 + -2x + -5 = 0 | 1. Identify coefficients: a=2, b=-2, c=-5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 4 - -40 = 44.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (2 ± √44) / 4.
5. Calculate roots: x1 = 2.1583123951777, x2 = -1.1583123951777.
Final Answer: x ... |
Compute the quotient and remainder of 88844 divided by 46. Show the long division steps. | To divide 88844 by 46:
1. Determine how many times 46 goes into the leading digits of 88844.
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: 88844 / 46 = 1931 with a remainder of 18.
Proof: (1931 * 46... |
Add the fractions 12/8 and 13/5. 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 12/8 by multiplying top and bottom by 5: 60/40.
3. Convert 13/5 by multiplying top and bottom by 8: 104/40.
4. Add the numerators: 60 + 104 = 164.
5. The resulting fraction is 164/40.
6. Simplify ... |
Find the roots of the quadratic equation: 10x^2 + -8x + -5 = 0 | 1. Identify coefficients: a=10, b=-8, c=-5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 64 - -200 = 264.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (8 ± √264) / 20.
5. Calculate roots: x1 = 1.2124038404635962, x2 = -0.41240384046359607.
Fin... |
Add the fractions 10/4 and 14/12. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 4 and 12, which is 12.
2. Convert 10/4 by multiplying top and bottom by 3: 30/12.
3. Convert 14/12 by multiplying top and bottom by 1: 14/12.
4. Add the numerators: 30 + 14 = 44.
5. The resulting fraction is 44/12.
6. Simplify by... |
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) |
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 |
Compute the quotient and remainder of 45320 divided by 93. Show the long division steps. | To divide 45320 by 93:
1. Determine how many times 93 goes into the leading digits of 45320.
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: 45320 / 93 = 487 with a remainder of 29.
Proof: (487 * 93) ... |
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 +... |
Add the fractions 10/3 and 6/13. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 3 and 13, which is 39.
2. Convert 10/3 by multiplying top and bottom by 13: 130/39.
3. Convert 6/13 by multiplying top and bottom by 3: 18/39.
4. Add the numerators: 130 + 18 = 148.
5. The resulting fraction is 148/39.
Final Answ... |
Multiply the following 2x2 matrices:
A =
[1, 1]
[1, 5]
B =
[5, -2]
[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) = (1*5) + (1*0) = 5
C12 = (A11 * B12) + (A12 * B22) = (1*-2) + (1*-5) = -7
C21 = (A21 * B11) + (A22 * B21) = (1*5) + (5*0) = 5
C22 = (A21 * B12) + (A22 * B22) = (1*-2) + (5*-5) = -27
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) |
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 |
In a right-angled triangle, the lengths of the two legs are a = 16 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: 16^2 + 8^2 = c^2
2. Calculate the squares: 256 + 64 = c^2
3. Sum the squares: 320 = c^2
4. Take the square root of both sides: c = √320
Final Ans... |
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 roots of the quadratic equation: 5x^2 + -7x + -1 = 0 | 1. Identify coefficients: a=5, b=-7, c=-1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 49 - -20 = 69.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (7 ± √69) / 10.
5. Calculate roots: x1 = 1.5306623862918074, x2 = -0.1306623862918075.
Final An... |
In a right-angled triangle, the lengths of the two legs are a = 16 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: 16^2 + 3^2 = c^2
2. Calculate the squares: 256 + 9 = c^2
3. Sum the squares: 265 = c^2
4. Take the square root of both sides: c = √265
Final Answ... |
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 |
In a right-angled triangle, the lengths of the two legs are a = 12 and b = 18. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 12^2 + 18^2 = c^2
2. Calculate the squares: 144 + 324 = c^2
3. Sum the squares: 468 = c^2
4. Take the square root of both sides: c = √468
Final A... |
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 |
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 |
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 =
[2, 2]
[2, -3]
B =
[-2, 5]
[1, 3] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (2*-2) + (2*1) = -2
C12 = (A11 * B12) + (A12 * B22) = (2*5) + (2*3) = 16
C21 = (A21 * B11) + (A22 * B21) = (2*-2) + (-3*1) = -7
C22 = (A21 * B12) + (A22 * B22) = (2*5) + (-3*3) = 1
Result Mat... |
Find the roots of the quadratic equation: 10x^2 + 7x + 0 = 0 | 1. Identify coefficients: a=10, b=7, c=0.
2. Calculate the discriminant: Δ = b^2 - 4ac = 49 - 0 = 49.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-7 ± √49) / 20.
5. Calculate roots: x1 = 0.0, x2 = -0.7.
Final Answer: x = 0.0, -0.7 |
Evaluate the logarithm: log_2(4) | To evaluate log_2(4), we ask the question: '2 raised to what power equals 4?'
Let x be the unknown power: 2^x = 4
Since 2^2 = 4, it follows that x = 2.
Final Answer: 2 |
Compute the quotient and remainder of 42933 divided by 19. Show the long division steps. | To divide 42933 by 19:
1. Determine how many times 19 goes into the leading digits of 42933.
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: 42933 / 19 = 2259 with a remainder of 12.
Proof: (2259 * 19... |
Find the derivative of f(x) = (3x + 4)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 3x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 4. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 4)^2 * 3 = 9(3x +... |
Evaluate the logarithm: log_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 |
Compute the indefinite integral of f(x) = 1x^4. | To find the indefinite integral ∫ 1x^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: 1 / 5 = 1/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 1x^4 dx = 1/5x^5 + C |
Find the derivative of f(x) = (2x + 5)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 2x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 2x + 5. Its derivative is h'(x) = 2.
4. Multiply g'(u) by h'(x): f'(x) = 3(2x + 5)^2 * 2 = 6(2x +... |
Multiply the following 2x2 matrices:
A =
[-5, 1]
[1, 1]
B =
[-4, 5]
[-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) = (-5*-4) + (1*-3) = 17
C12 = (A11 * B12) + (A12 * B22) = (-5*5) + (1*-1) = -26
C21 = (A21 * B11) + (A22 * B21) = (1*-4) + (1*-3) = -7
C22 = (A21 * B12) + (A22 * B22) = (1*5) + (1*-1) = 4
Resul... |
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 |
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) |
Calculate the mean, population variance, and standard deviation for the dataset: [9, 4, 7, 13, 11] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (9 + 4 + 7 + 13 + 11) / 5 = 44 / 5 = 8.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [0.2, -4.8, -1.8, 4.2, 2.2]
Squared Differences: [0.04, 23.04, 3.24, 17.64, 4.84]
S... |
In a right-angled triangle, the lengths of the two legs are a = 3 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: 3^2 + 11^2 = c^2
2. Calculate the squares: 9 + 121 = c^2
3. Sum the squares: 130 = c^2
4. Take the square root of both sides: c = √130
Final Answ... |
Find the roots of the quadratic equation: 6x^2 + -5x + -7 = 0 | 1. Identify coefficients: a=6, b=-5, c=-7.
2. Calculate the discriminant: Δ = b^2 - 4ac = 25 - -168 = 193.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (5 ± √193) / 12.
5. Calculate roots: x1 = 1.5743703324541503, x2 = -0.741036999120817.
Final ... |
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) |
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 +... |
Add the fractions 11/4 and 1/5. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 4 and 5, which is 20.
2. Convert 11/4 by multiplying top and bottom by 5: 55/20.
3. Convert 1/5 by multiplying top and bottom by 4: 4/20.
4. Add the numerators: 55 + 4 = 59.
5. The resulting fraction is 59/20.
Final Answer: 59/20 |
Find the roots of the quadratic equation: 2x^2 + 1x + 4 = 0 | 1. Identify coefficients: a=2, b=1, c=4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1^2 - 4(2)(4) = 1 - 32 = -31.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-1 ± √31i) / 4.
Final Answer: x = -0.25 ± 1.3919410907075054i |
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