prompt stringlengths 32 115 | completion stringlengths 167 566 |
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Multiply the following 2x2 matrices:
A =
[-4, 5]
[-5, 3]
B =
[-5, 0]
[-5, -2] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-4*-5) + (5*-5) = -5
C12 = (A11 * B12) + (A12 * B22) = (-4*0) + (5*-2) = -10
C21 = (A21 * B11) + (A22 * B21) = (-5*-5) + (3*-5) = 10
C22 = (A21 * B12) + (A22 * B22) = (-5*0) + (3*-2) = -6
Re... |
Compute the quotient and remainder of 74865 divided by 89. Show the long division steps. | To divide 74865 by 89:
1. Determine how many times 89 goes into the leading digits of 74865.
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: 74865 / 89 = 841 with a remainder of 16.
Proof: (841 * 89) ... |
Add the fractions 8/3 and 7/6. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 3 and 6, which is 6.
2. Convert 8/3 by multiplying top and bottom by 2: 16/6.
3. Convert 7/6 by multiplying top and bottom by 1: 7/6.
4. Add the numerators: 16 + 7 = 23.
5. The resulting fraction is 23/6.
Final Answer: 23/6 |
Calculate the mean, population variance, and standard deviation for the dataset: [15, 18, 3, 10, 4] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (15 + 18 + 3 + 10 + 4) / 5 = 50 / 5 = 10.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [5.0, 8.0, -7.0, 0.0, -6.0]
Squared Differences: [25.0, 64.0, 49.0, 0.0, 36.0]
Su... |
Add the fractions 14/3 and 12/7. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 3 and 7, which is 21.
2. Convert 14/3 by multiplying top and bottom by 7: 98/21.
3. Convert 12/7 by multiplying top and bottom by 3: 36/21.
4. Add the numerators: 98 + 36 = 134.
5. The resulting fraction is 134/21.
Final Answer: ... |
Calculate the mean, population variance, and standard deviation for the dataset: [17, 11, 6, 18, 19] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (17 + 11 + 6 + 18 + 19) / 5 = 71 / 5 = 14.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [2.8, -3.2, -8.2, 3.8, 4.8]
Squared Differences: [7.84, 10.24, 67.24, 14.44, 23.04]... |
Find the derivative of f(x) = (4x + 4)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 4x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 4. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 4)^2 * 4 = 12(4x ... |
Compute the indefinite integral of f(x) = 10x^3. | To find the indefinite integral ∫ 10x^3 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 3 + 1 = 4.
2. Divide the coefficient by the new exponent: 10 / 4 = 5/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 10x^3 dx = 5/2x^4 + C |
Calculate the mean, population variance, and standard deviation for the dataset: [4, 9, 17, 13, 9] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (4 + 9 + 17 + 13 + 9) / 5 = 52 / 5 = 10.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-6.4, -1.4, 6.6, 2.6, -1.4]
Squared Differences: [40.96, 1.96, 43.56, 6.76, 1.96]
... |
Compute the indefinite integral of f(x) = 6x^5. | To find the indefinite integral ∫ 6x^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: 6 / 6 = 1.
3. Combine and add the constant of integration C.
Final Answer: ∫ 6x^5 dx = 1x^6 + C |
Multiply the following 2x2 matrices:
A =
[-5, -5]
[5, -2]
B =
[-1, -5]
[-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) + (-5*-5) = 30
C12 = (A11 * B12) + (A12 * B22) = (-5*-5) + (-5*4) = 5
C21 = (A21 * B11) + (A22 * B21) = (5*-1) + (-2*-5) = 5
C22 = (A21 * B12) + (A22 * B22) = (5*-5) + (-2*4) = -33
Re... |
Find the roots of the quadratic equation: 5x^2 + 6x + 4 = 0 | 1. Identify coefficients: a=5, b=6, c=4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 6^2 - 4(5)(4) = 36 - 80 = -44.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-6 ± √44i) / 10.
Final Answer: x = -0.6 ± 0.6633249580710799i |
Multiply the following 2x2 matrices:
A =
[5, 3]
[-4, 2]
B =
[4, -4]
[5, -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*4) + (3*5) = 35
C12 = (A11 * B12) + (A12 * B22) = (5*-4) + (3*-3) = -29
C21 = (A21 * B11) + (A22 * B21) = (-4*4) + (2*5) = -6
C22 = (A21 * B12) + (A22 * B22) = (-4*-4) + (2*-3) = 10
Result... |
Compute the quotient and remainder of 22951 divided by 92. Show the long division steps. | To divide 22951 by 92:
1. Determine how many times 92 goes into the leading digits of 22951.
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: 22951 / 92 = 249 with a remainder of 43.
Proof: (249 * 92) ... |
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 |
Compute the indefinite integral of f(x) = 10x^2. | To find the indefinite integral ∫ 10x^2 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 2 + 1 = 3.
2. Divide the coefficient by the new exponent: 10 / 3 = 10/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 10x^2 dx = 10/3x^3 + C |
Compute the quotient and remainder of 55000 divided by 93. Show the long division steps. | To divide 55000 by 93:
1. Determine how many times 93 goes into the leading digits of 55000.
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: 55000 / 93 = 591 with a remainder of 37.
Proof: (591 * 93) ... |
Find the roots of the quadratic equation: 1x^2 + 9x + -4 = 0 | 1. Identify coefficients: a=1, b=9, c=-4.
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) / 2.
5. Calculate roots: x1 = 0.42442890089805196, x2 = -9.424428900898052.
Final Ans... |
Compute the indefinite integral of f(x) = 4x^2. | To find the indefinite integral ∫ 4x^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: 4 / 3 = 4/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 4x^2 dx = 4/3x^3 + C |
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 +... |
Multiply the following 2x2 matrices:
A =
[2, -1]
[0, -5]
B =
[-4, 1]
[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) = (2*-4) + (-1*3) = -11
C12 = (A11 * B12) + (A12 * B22) = (2*1) + (-1*4) = -2
C21 = (A21 * B11) + (A22 * B21) = (0*-4) + (-5*3) = -15
C22 = (A21 * B12) + (A22 * B22) = (0*1) + (-5*4) = -20
Resu... |
Compute the quotient and remainder of 84869 divided by 85. Show the long division steps. | To divide 84869 by 85:
1. Determine how many times 85 goes into the leading digits of 84869.
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: 84869 / 85 = 998 with a remainder of 39.
Proof: (998 * 85) ... |
Add the fractions 13/2 and 14/9. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 2 and 9, which is 18.
2. Convert 13/2 by multiplying top and bottom by 9: 117/18.
3. Convert 14/9 by multiplying top and bottom by 2: 28/18.
4. Add the numerators: 117 + 28 = 145.
5. The resulting fraction is 145/18.
Final Answer... |
In a right-angled triangle, the lengths of the two legs are a = 16 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: 16^2 + 19^2 = c^2
2. Calculate the squares: 256 + 361 = c^2
3. Sum the squares: 617 = c^2
4. Take the square root of both sides: c = √617
Final A... |
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 ... |
Find the roots of the quadratic equation: 4x^2 + -2x + -5 = 0 | 1. Identify coefficients: a=4, b=-2, c=-5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 4 - -80 = 84.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (2 ± √84) / 8.
5. Calculate roots: x1 = 1.39564392373896, x2 = -0.89564392373896.
Final Answer: ... |
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 ... |
Add the fractions 9/8 and 9/8. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 8 and 8, which is 8.
2. Convert 9/8 by multiplying top and bottom by 1: 9/8.
3. Convert 9/8 by multiplying top and bottom by 1: 9/8.
4. Add the numerators: 9 + 9 = 18.
5. The resulting fraction is 18/8.
6. Simplify by dividing nu... |
Compute the quotient and remainder of 62262 divided by 16. Show the long division steps. | To divide 62262 by 16:
1. Determine how many times 16 goes into the leading digits of 62262.
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: 62262 / 16 = 3891 with a remainder of 6.
Proof: (3891 * 16)... |
Add the fractions 14/14 and 9/12. 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 14/14 by multiplying top and bottom by 6: 84/84.
3. Convert 9/12 by multiplying top and bottom by 7: 63/84.
4. Add the numerators: 84 + 63 = 147.
5. The resulting fraction is 147/84.
6. Simplify... |
In a right-angled triangle, the lengths of the two legs are a = 19 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: 19^2 + 12^2 = c^2
2. Calculate the squares: 361 + 144 = c^2
3. Sum the squares: 505 = c^2
4. Take the square root of both sides: c = √505
Final A... |
Calculate the mean, population variance, and standard deviation for the dataset: [12, 12, 19, 6, 20] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (12 + 12 + 19 + 6 + 20) / 5 = 69 / 5 = 13.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-1.8, -1.8, 5.2, -7.8, 6.2]
Squared Differences: [3.24, 3.24, 27.04, 60.84, 38.44]... |
Compute the quotient and remainder of 4104 divided by 25. Show the long division steps. | To divide 4104 by 25:
1. Determine how many times 25 goes into the leading digits of 4104.
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: 4104 / 25 = 164 with a remainder of 4.
Proof: (164 * 25) + 4 ... |
Find the roots of the quadratic equation: 3x^2 + -6x + -4 = 0 | 1. Identify coefficients: a=3, b=-6, c=-4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 36 - -48 = 84.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (6 ± √84) / 6.
5. Calculate roots: x1 = 2.5275252316519468, x2 = -0.5275252316519466.
Final Ans... |
Calculate the mean, population variance, and standard deviation for the dataset: [1, 9, 7, 20, 3] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (1 + 9 + 7 + 20 + 3) / 5 = 40 / 5 = 8.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-7.0, 1.0, -1.0, 12.0, -5.0]
Squared Differences: [49.0, 1.0, 1.0, 144.0, 25.0]
Sum... |
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 |
Add the fractions 1/4 and 2/2. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 4 and 2, which is 4.
2. Convert 1/4 by multiplying top and bottom by 1: 1/4.
3. Convert 2/2 by multiplying top and bottom by 2: 4/4.
4. Add the numerators: 1 + 4 = 5.
5. The resulting fraction is 5/4.
Final Answer: 5/4 |
Compute the quotient and remainder of 18357 divided by 53. Show the long division steps. | To divide 18357 by 53:
1. Determine how many times 53 goes into the leading digits of 18357.
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: 18357 / 53 = 346 with a remainder of 19.
Proof: (346 * 53) ... |
Compute the quotient and remainder of 91455 divided by 81. Show the long division steps. | To divide 91455 by 81:
1. Determine how many times 81 goes into the leading digits of 91455.
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: 91455 / 81 = 1129 with a remainder of 6.
Proof: (1129 * 81)... |
In a right-angled triangle, the lengths of the two legs are a = 19 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: 19^2 + 12^2 = c^2
2. Calculate the squares: 361 + 144 = c^2
3. Sum the squares: 505 = c^2
4. Take the square root of both sides: c = √505
Final A... |
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 |
Find the roots of the quadratic equation: 8x^2 + -3x + -5 = 0 | 1. Identify coefficients: a=8, b=-3, c=-5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 9 - -160 = 169.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (3 ± √169) / 16.
5. Calculate roots: x1 = 1.0, x2 = -0.625.
Final Answer: x = 1.0, -0.625 |
Add the fractions 3/6 and 9/6. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 6 and 6, which is 6.
2. Convert 3/6 by multiplying top and bottom by 1: 3/6.
3. Convert 9/6 by multiplying top and bottom by 1: 9/6.
4. Add the numerators: 3 + 9 = 12.
5. The resulting fraction is 12/6.
6. Simplify by dividing nu... |
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 +... |
In a right-angled triangle, the lengths of the two legs are a = 10 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: 10^2 + 12^2 = c^2
2. Calculate the squares: 100 + 144 = c^2
3. Sum the squares: 244 = c^2
4. Take the square root of both sides: c = √244
Final A... |
Calculate the mean, population variance, and standard deviation for the dataset: [11, 18, 8, 6, 8] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (11 + 18 + 8 + 6 + 8) / 5 = 51 / 5 = 10.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [0.8, 7.8, -2.2, -4.2, -2.2]
Squared Differences: [0.64, 60.84, 4.84, 17.64, 4.84]
... |
Multiply the following 2x2 matrices:
A =
[1, 5]
[4, 3]
B =
[-5, 4]
[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*-5) + (5*3) = 10
C12 = (A11 * B12) + (A12 * B22) = (1*4) + (5*3) = 19
C21 = (A21 * B11) + (A22 * B21) = (4*-5) + (3*3) = -11
C22 = (A21 * B12) + (A22 * B22) = (4*4) + (3*3) = 25
Result Mat... |
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 |
Multiply the following 2x2 matrices:
A =
[-1, -5]
[-5, -1]
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) = (-1*3) + (-5*2) = -13
C12 = (A11 * B12) + (A12 * B22) = (-1*-2) + (-5*-3) = 17
C21 = (A21 * B11) + (A22 * B21) = (-5*3) + (-1*2) = -17
C22 = (A21 * B12) + (A22 * B22) = (-5*-2) + (-1*-3) = 13
... |
Compute the quotient and remainder of 99060 divided by 67. Show the long division steps. | To divide 99060 by 67:
1. Determine how many times 67 goes into the leading digits of 99060.
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: 99060 / 67 = 1478 with a remainder of 34.
Proof: (1478 * 67... |
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) |
Add the fractions 14/2 and 8/4. 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 14/2 by multiplying top and bottom by 2: 28/4.
3. Convert 8/4 by multiplying top and bottom by 1: 8/4.
4. Add the numerators: 28 + 8 = 36.
5. The resulting fraction is 36/4.
6. Simplify by dividing... |
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 |
Add the fractions 5/9 and 4/9. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 9 and 9, which is 9.
2. Convert 5/9 by multiplying top and bottom by 1: 5/9.
3. Convert 4/9 by multiplying top and bottom by 1: 4/9.
4. Add the numerators: 5 + 4 = 9.
5. The resulting fraction is 9/9.
6. Simplify by dividing nume... |
Compute the quotient and remainder of 33097 divided by 45. Show the long division steps. | To divide 33097 by 45:
1. Determine how many times 45 goes into the leading digits of 33097.
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: 33097 / 45 = 735 with a remainder of 22.
Proof: (735 * 45) ... |
Evaluate cos(45°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(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_4(1024) | To evaluate log_4(1024), we ask the question: '4 raised to what power equals 1024?'
Let x be the unknown power: 4^x = 1024
Since 4^5 = 1024, it follows that x = 5.
Final Answer: 5 |
Add the fractions 5/6 and 9/10. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 6 and 10, which is 30.
2. Convert 5/6 by multiplying top and bottom by 5: 25/30.
3. Convert 9/10 by multiplying top and bottom by 3: 27/30.
4. Add the numerators: 25 + 27 = 52.
5. The resulting fraction is 52/30.
6. Simplify by d... |
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 +... |
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 |
Calculate the mean, population variance, and standard deviation for the dataset: [8, 7, 7, 20, 16] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (8 + 7 + 7 + 20 + 16) / 5 = 58 / 5 = 11.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-3.6, -4.6, -4.6, 8.4, 4.4]
Squared Differences: [12.96, 21.16, 21.16, 70.56, 19.36]... |
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: 1x^2 + -10x + 1 = 0 | 1. Identify coefficients: a=1, b=-10, c=1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 100 - 4 = 96.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (10 ± √96) / 2.
5. Calculate roots: x1 = 9.898979485566356, x2 = 0.10102051443364424.
Final Answ... |
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 |
Compute the indefinite integral of f(x) = 10x^3. | To find the indefinite integral ∫ 10x^3 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 3 + 1 = 4.
2. Divide the coefficient by the new exponent: 10 / 4 = 5/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 10x^3 dx = 5/2x^4 + C |
Find the 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 quotient and remainder of 43142 divided by 88. Show the long division steps. | To divide 43142 by 88:
1. Determine how many times 88 goes into the leading digits of 43142.
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: 43142 / 88 = 490 with a remainder of 22.
Proof: (490 * 88) ... |
Find the derivative of f(x) = (4x + 5)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 4x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 5. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 5)^2 * 4 = 12(4x ... |
Compute the indefinite integral of f(x) = 2x^3. | To find the indefinite integral ∫ 2x^3 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 3 + 1 = 4.
2. Divide the coefficient by the new exponent: 2 / 4 = 1/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 2x^3 dx = 1/2x^4 + C |
Multiply the following 2x2 matrices:
A =
[-5, 1]
[1, 4]
B =
[4, -4]
[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*-4) + (1*-5) = 15
C21 = (A21 * B11) + (A22 * B21) = (1*4) + (4*0) = 4
C22 = (A21 * B12) + (A22 * B22) = (1*-4) + (4*-5) = -24
Result... |
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 +... |
Multiply the following 2x2 matrices:
A =
[0, 3]
[-2, 3]
B =
[0, 2]
[-1, -1] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (0*0) + (3*-1) = -3
C12 = (A11 * B12) + (A12 * B22) = (0*2) + (3*-1) = -3
C21 = (A21 * B11) + (A22 * B21) = (-2*0) + (3*-1) = -3
C22 = (A21 * B12) + (A22 * B22) = (-2*2) + (3*-1) = -7
Result ... |
In a right-angled triangle, the lengths of the two legs are a = 5 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: 5^2 + 15^2 = c^2
2. Calculate the squares: 25 + 225 = c^2
3. Sum the squares: 250 = c^2
4. Take the square root of both sides: c = √250
Final Ans... |
Add the fractions 10/15 and 10/6. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 15 and 6, which is 30.
2. Convert 10/15 by multiplying top and bottom by 2: 20/30.
3. Convert 10/6 by multiplying top and bottom by 5: 50/30.
4. Add the numerators: 20 + 50 = 70.
5. The resulting fraction is 70/30.
6. Simplify by... |
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 7534 divided by 79. Show the long division steps. | To divide 7534 by 79:
1. Determine how many times 79 goes into the leading digits of 7534.
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: 7534 / 79 = 95 with a remainder of 29.
Proof: (95 * 79) + 29 ... |
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) |
Multiply the following 2x2 matrices:
A =
[-4, 2]
[1, 2]
B =
[-3, -5]
[-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) = (-4*-3) + (2*-3) = 6
C12 = (A11 * B12) + (A12 * B22) = (-4*-5) + (2*-4) = 12
C21 = (A21 * B11) + (A22 * B21) = (1*-3) + (2*-3) = -9
C22 = (A21 * B12) + (A22 * B22) = (1*-5) + (2*-4) = -13
Res... |
In a right-angled triangle, the lengths of the two legs are a = 10 and b = 20. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 10^2 + 20^2 = c^2
2. Calculate the squares: 100 + 400 = c^2
3. Sum the squares: 500 = c^2
4. Take the square root of both sides: c = √500
Final A... |
Calculate the mean, population variance, and standard deviation for the dataset: [19, 7, 12, 11, 15] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (19 + 7 + 12 + 11 + 15) / 5 = 64 / 5 = 12.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [6.2, -5.8, -0.8, -1.8, 2.2]
Squared Differences: [38.44, 33.64, 0.64, 3.24, 4.84]
... |
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 |
Calculate the mean, population variance, and standard deviation for the dataset: [20, 2, 10, 3, 11] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (20 + 2 + 10 + 3 + 11) / 5 = 46 / 5 = 9.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [10.8, -7.2, 0.8, -6.2, 1.8]
Squared Differences: [116.64, 51.84, 0.64, 38.44, 3.24]
... |
In a right-angled triangle, the lengths of the two legs are a = 3 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: 3^2 + 10^2 = c^2
2. Calculate the squares: 9 + 100 = c^2
3. Sum the squares: 109 = c^2
4. Take the square root of both sides: c = √109
Final Answ... |
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: [6, 18, 7, 14, 11] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (6 + 18 + 7 + 14 + 11) / 5 = 56 / 5 = 11.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-5.2, 6.8, -4.2, 2.8, -0.2]
Squared Differences: [27.04, 46.24, 17.64, 7.84, 0.04]
... |
Multiply the following 2x2 matrices:
A =
[2, 1]
[0, 0]
B =
[5, 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*5) + (1*1) = 11
C12 = (A11 * B12) + (A12 * B22) = (2*1) + (1*-2) = 0
C21 = (A21 * B11) + (A22 * B21) = (0*5) + (0*1) = 0
C22 = (A21 * B12) + (A22 * B22) = (0*1) + (0*-2) = 0
Result Matrix ... |
In a right-angled triangle, the lengths of the two legs are a = 18 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: 18^2 + 4^2 = c^2
2. Calculate the squares: 324 + 16 = c^2
3. Sum the squares: 340 = c^2
4. Take the square root of both sides: c = √340
Final Ans... |
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 9/14 and 3/6. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 14 and 6, which is 42.
2. Convert 9/14 by multiplying top and bottom by 3: 27/42.
3. Convert 3/6 by multiplying top and bottom by 7: 21/42.
4. Add the numerators: 27 + 21 = 48.
5. The resulting fraction is 48/42.
6. Simplify by d... |
Find the roots of the quadratic equation: 4x^2 + 7x + -6 = 0 | 1. Identify coefficients: a=4, b=7, c=-6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 49 - -96 = 145.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-7 ± √145) / 8.
5. Calculate roots: x1 = 0.630199322349037, x2 = -2.380199322349037.
Final Ans... |
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 |
In a right-angled triangle, the lengths of the two legs are a = 8 and b = 16. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 8^2 + 16^2 = c^2
2. Calculate the squares: 64 + 256 = c^2
3. Sum the squares: 320 = c^2
4. Take the square root of both sides: c = √320
Final Ans... |
Compute the indefinite integral of f(x) = 6x^2. | To find the indefinite integral ∫ 6x^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: 6 / 3 = 2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 6x^2 dx = 2x^3 + C |
Compute the quotient and remainder of 67555 divided by 40. Show the long division steps. | To divide 67555 by 40:
1. Determine how many times 40 goes into the leading digits of 67555.
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: 67555 / 40 = 1688 with a remainder of 35.
Proof: (1688 * 40... |
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) |
Calculate the mean, population variance, and standard deviation for the dataset: [14, 7, 18, 17, 11] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (14 + 7 + 18 + 17 + 11) / 5 = 67 / 5 = 13.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [0.6, -6.4, 4.6, 3.6, -2.4]
Squared Differences: [0.36, 40.96, 21.16, 12.96, 5.76]
... |
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 |
Multiply the following 2x2 matrices:
A =
[-4, 3]
[-2, -3]
B =
[-4, 0]
[1, -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*-4) + (3*1) = 19
C12 = (A11 * B12) + (A12 * B22) = (-4*0) + (3*-5) = -15
C21 = (A21 * B11) + (A22 * B21) = (-2*-4) + (-3*1) = 5
C22 = (A21 * B12) + (A22 * B22) = (-2*0) + (-3*-5) = 15
Res... |
Calculate the mean, population variance, and standard deviation for the dataset: [5, 7, 13, 2, 4] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (5 + 7 + 13 + 2 + 4) / 5 = 31 / 5 = 6.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-1.2, 0.8, 6.8, -4.2, -2.2]
Squared Differences: [1.44, 0.64, 46.24, 17.64, 4.84]
S... |
Compute the quotient and remainder of 79721 divided by 27. Show the long division steps. | To divide 79721 by 27:
1. Determine how many times 27 goes into the leading digits of 79721.
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: 79721 / 27 = 2952 with a remainder of 17.
Proof: (2952 * 27... |
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