prompt stringlengths 32 115 | completion stringlengths 167 566 |
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Multiply the following 2x2 matrices:
A =
[-3, 4]
[5, 1]
B =
[4, -1]
[-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) = (-3*4) + (4*-2) = -20
C12 = (A11 * B12) + (A12 * B22) = (-3*-1) + (4*2) = 11
C21 = (A21 * B11) + (A22 * B21) = (5*4) + (1*-2) = 18
C22 = (A21 * B12) + (A22 * B22) = (5*-1) + (1*2) = -3
Result... |
In a right-angled triangle, the lengths of the two legs are a = 16 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: 16^2 + 15^2 = c^2
2. Calculate the squares: 256 + 225 = c^2
3. Sum the squares: 481 = c^2
4. Take the square root of both sides: c = √481
Final A... |
Calculate the mean, population variance, and standard deviation for the dataset: [6, 15, 1, 16, 3] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (6 + 15 + 1 + 16 + 3) / 5 = 41 / 5 = 8.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-2.2, 6.8, -7.2, 7.8, -5.2]
Squared Differences: [4.84, 46.24, 51.84, 60.84, 27.04]
... |
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 |
Compute the quotient and remainder of 57519 divided by 84. Show the long division steps. | To divide 57519 by 84:
1. Determine how many times 84 goes into the leading digits of 57519.
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: 57519 / 84 = 684 with a remainder of 63.
Proof: (684 * 84) ... |
Calculate the mean, population variance, and standard deviation for the dataset: [10, 11, 12, 10, 20] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (10 + 11 + 12 + 10 + 20) / 5 = 63 / 5 = 12.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-2.6, -1.6, -0.6, -2.6, 7.4]
Squared Differences: [6.76, 2.56, 0.36, 6.76, 54.76]... |
Compute the quotient and remainder of 79910 divided by 34. Show the long division steps. | To divide 79910 by 34:
1. Determine how many times 34 goes into the leading digits of 79910.
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: 79910 / 34 = 2350 with a remainder of 10.
Proof: (2350 * 34... |
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 |
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 |
Multiply the following 2x2 matrices:
A =
[-3, -1]
[1, -4]
B =
[0, 2]
[4, -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*0) + (-1*4) = -4
C12 = (A11 * B12) + (A12 * B22) = (-3*2) + (-1*-4) = -2
C21 = (A21 * B11) + (A22 * B21) = (1*0) + (-4*4) = -16
C22 = (A21 * B12) + (A22 * B22) = (1*2) + (-4*-4) = 18
Resu... |
Add the fractions 12/3 and 6/8. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 3 and 8, which is 24.
2. Convert 12/3 by multiplying top and bottom by 8: 96/24.
3. Convert 6/8 by multiplying top and bottom by 3: 18/24.
4. Add the numerators: 96 + 18 = 114.
5. The resulting fraction is 114/24.
6. Simplify by ... |
In a right-angled triangle, the lengths of the two legs are a = 19 and b = 6. 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 + 6^2 = c^2
2. Calculate the squares: 361 + 36 = c^2
3. Sum the squares: 397 = c^2
4. Take the square root of both sides: c = √397
Final Ans... |
Compute the indefinite integral of f(x) = 3x^1. | To find the indefinite integral ∫ 3x^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: 3 / 2 = 3/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 3x^1 dx = 3/2x^2 + C |
Evaluate the logarithm: log_3(243) | To evaluate log_3(243), we ask the question: '3 raised to what power equals 243?'
Let x be the unknown power: 3^x = 243
Since 3^5 = 243, it follows that x = 5.
Final Answer: 5 |
Compute the quotient and remainder of 60312 divided by 83. Show the long division steps. | To divide 60312 by 83:
1. Determine how many times 83 goes into the leading digits of 60312.
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: 60312 / 83 = 726 with a remainder of 54.
Proof: (726 * 83) ... |
Evaluate the logarithm: log_3(243) | To evaluate log_3(243), we ask the question: '3 raised to what power equals 243?'
Let x be the unknown power: 3^x = 243
Since 3^5 = 243, it follows that x = 5.
Final Answer: 5 |
Find the 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 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) |
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 +... |
Compute the quotient and remainder of 73402 divided by 67. Show the long division steps. | To divide 73402 by 67:
1. Determine how many times 67 goes into the leading digits of 73402.
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: 73402 / 67 = 1095 with a remainder of 37.
Proof: (1095 * 67... |
Find the roots of the quadratic equation: 5x^2 + -2x + -5 = 0 | 1. Identify coefficients: a=5, b=-2, c=-5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 4 - -100 = 104.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (2 ± √104) / 10.
5. Calculate roots: x1 = 1.219803902718557, x2 = -0.8198039027185569.
Final A... |
Add the fractions 5/15 and 14/13. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 15 and 13, which is 195.
2. Convert 5/15 by multiplying top and bottom by 13: 65/195.
3. Convert 14/13 by multiplying top and bottom by 15: 210/195.
4. Add the numerators: 65 + 210 = 275.
5. The resulting fraction is 275/195.
6. ... |
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 +... |
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) |
Add the fractions 10/10 and 15/15. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 10 and 15, which is 30.
2. Convert 10/10 by multiplying top and bottom by 3: 30/30.
3. Convert 15/15 by multiplying top and bottom by 2: 30/30.
4. Add the numerators: 30 + 30 = 60.
5. The resulting fraction is 60/30.
6. Simplify ... |
Calculate the mean, population variance, and standard deviation for the dataset: [12, 4, 7, 17, 17] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (12 + 4 + 7 + 17 + 17) / 5 = 57 / 5 = 11.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [0.6, -7.4, -4.4, 5.6, 5.6]
Squared Differences: [0.36, 54.76, 19.36, 31.36, 31.36]
... |
Compute the quotient and remainder of 49359 divided by 9. Show the long division steps. | To divide 49359 by 9:
1. Determine how many times 9 goes into the leading digits of 49359.
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: 49359 / 9 = 5484 with a remainder of 3.
Proof: (5484 * 9) + 3... |
Add the fractions 8/3 and 11/9. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 3 and 9, which is 9.
2. Convert 8/3 by multiplying top and bottom by 3: 24/9.
3. Convert 11/9 by multiplying top and bottom by 1: 11/9.
4. Add the numerators: 24 + 11 = 35.
5. The resulting fraction is 35/9.
Final Answer: 35/9 |
Compute the indefinite integral of f(x) = 5x^1. | To find the indefinite integral ∫ 5x^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: 5 / 2 = 5/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 5x^1 dx = 5/2x^2 + C |
Multiply the following 2x2 matrices:
A =
[-5, -1]
[-1, -5]
B =
[5, 2]
[5, -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*5) + (-1*5) = -30
C12 = (A11 * B12) + (A12 * B22) = (-5*2) + (-1*-5) = -5
C21 = (A21 * B11) + (A22 * B21) = (-1*5) + (-5*5) = -30
C22 = (A21 * B12) + (A22 * B22) = (-1*2) + (-5*-5) = 23
R... |
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) |
Find the roots of the quadratic equation: 1x^2 + 9x + -6 = 0 | 1. Identify coefficients: a=1, b=9, c=-6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 81 - -24 = 105.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-9 ± √105) / 2.
5. Calculate roots: x1 = 0.623475382979799, x2 = -9.623475382979798.
Final Ans... |
Compute the quotient and remainder of 80019 divided by 93. Show the long division steps. | To divide 80019 by 93:
1. Determine how many times 93 goes into the leading digits of 80019.
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: 80019 / 93 = 860 with a remainder of 39.
Proof: (860 * 93) ... |
Compute the indefinite integral of f(x) = 5x^1. | To find the indefinite integral ∫ 5x^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: 5 / 2 = 5/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 5x^1 dx = 5/2x^2 + C |
Compute the indefinite integral of f(x) = 6x^4. | To find the indefinite integral ∫ 6x^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: 6 / 5 = 6/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 6x^4 dx = 6/5x^5 + C |
Multiply the following 2x2 matrices:
A =
[-1, 4]
[5, 1]
B =
[1, 5]
[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) = (-1*1) + (4*1) = 3
C12 = (A11 * B12) + (A12 * B22) = (-1*5) + (4*5) = 15
C21 = (A21 * B11) + (A22 * B21) = (5*1) + (1*1) = 6
C22 = (A21 * B12) + (A22 * B22) = (5*5) + (1*5) = 30
Result Matrix... |
Evaluate cos(30°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(30°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.8660 (exact form depends on root simplification if applicable) |
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) |
Compute the quotient and remainder of 51141 divided by 63. Show the long division steps. | To divide 51141 by 63:
1. Determine how many times 63 goes into the leading digits of 51141.
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: 51141 / 63 = 811 with a remainder of 48.
Proof: (811 * 63) ... |
Add the fractions 10/14 and 11/11. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 14 and 11, which is 154.
2. Convert 10/14 by multiplying top and bottom by 11: 110/154.
3. Convert 11/11 by multiplying top and bottom by 14: 154/154.
4. Add the numerators: 110 + 154 = 264.
5. The resulting fraction is 264/154.
... |
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 7/9 and 10/11. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 9 and 11, which is 99.
2. Convert 7/9 by multiplying top and bottom by 11: 77/99.
3. Convert 10/11 by multiplying top and bottom by 9: 90/99.
4. Add the numerators: 77 + 90 = 167.
5. The resulting fraction is 167/99.
Final Answer... |
Multiply the following 2x2 matrices:
A =
[-1, -2]
[-2, 2]
B =
[-4, 1]
[-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) = (-1*-4) + (-2*-2) = 8
C12 = (A11 * B12) + (A12 * B22) = (-1*1) + (-2*-5) = 9
C21 = (A21 * B11) + (A22 * B21) = (-2*-4) + (2*-2) = 4
C22 = (A21 * B12) + (A22 * B22) = (-2*1) + (2*-5) = -12
Res... |
In a right-angled triangle, the lengths of the two legs are a = 13 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: 13^2 + 19^2 = c^2
2. Calculate the squares: 169 + 361 = c^2
3. Sum the squares: 530 = c^2
4. Take the square root of both sides: c = √530
Final A... |
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) |
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 +... |
Find the roots of the quadratic equation: 4x^2 + 0x + -5 = 0 | 1. Identify coefficients: a=4, b=0, c=-5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 0 - -80 = 80.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (0 ± √80) / 8.
5. Calculate roots: x1 = 1.118033988749895, x2 = -1.118033988749895.
Final Answer:... |
In a right-angled triangle, the lengths of the two legs are a = 5 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: 5^2 + 12^2 = c^2
2. Calculate the squares: 25 + 144 = c^2
3. Sum the squares: 169 = c^2
4. Take the square root of both sides: c = √169
Final Ans... |
Evaluate the logarithm: log_5(25) | To evaluate log_5(25), we ask the question: '5 raised to what power equals 25?'
Let x be the unknown power: 5^x = 25
Since 5^2 = 25, it follows that x = 2.
Final Answer: 2 |
Compute the indefinite integral of f(x) = 2x^5. | To find the indefinite integral ∫ 2x^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: 2 / 6 = 1/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 2x^5 dx = 1/3x^6 + C |
Add the fractions 3/7 and 14/10. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 7 and 10, which is 70.
2. Convert 3/7 by multiplying top and bottom by 10: 30/70.
3. Convert 14/10 by multiplying top and bottom by 7: 98/70.
4. Add the numerators: 30 + 98 = 128.
5. The resulting fraction is 128/70.
6. Simplify ... |
Compute the quotient and remainder of 55205 divided by 15. Show the long division steps. | To divide 55205 by 15:
1. Determine how many times 15 goes into the leading digits of 55205.
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: 55205 / 15 = 3680 with a remainder of 5.
Proof: (3680 * 15)... |
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 roots of the quadratic equation: 1x^2 + 4x + -10 = 0 | 1. Identify coefficients: a=1, b=4, c=-10.
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) / 2.
5. Calculate roots: x1 = 1.7416573867739413, x2 = -5.741657386773941.
Final Ans... |
In a right-angled triangle, the lengths of the two legs are a = 14 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: 14^2 + 11^2 = c^2
2. Calculate the squares: 196 + 121 = c^2
3. Sum the squares: 317 = c^2
4. Take the square root of both sides: c = √317
Final A... |
Add the fractions 15/5 and 15/5. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 5 and 5, which is 5.
2. Convert 15/5 by multiplying top and bottom by 1: 15/5.
3. Convert 15/5 by multiplying top and bottom by 1: 15/5.
4. Add the numerators: 15 + 15 = 30.
5. The resulting fraction is 30/5.
6. Simplify by divid... |
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 ... |
Calculate the mean, population variance, and standard deviation for the dataset: [9, 14, 7, 14, 14] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (9 + 14 + 7 + 14 + 14) / 5 = 58 / 5 = 11.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-2.6, 2.4, -4.6, 2.4, 2.4]
Squared Differences: [6.76, 5.76, 21.16, 5.76, 5.76]
... |
Calculate the mean, population variance, and standard deviation for the dataset: [3, 5, 4, 14, 14] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (3 + 5 + 4 + 14 + 14) / 5 = 40 / 5 = 8.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-5.0, -3.0, -4.0, 6.0, 6.0]
Squared Differences: [25.0, 9.0, 16.0, 36.0, 36.0]
Sum... |
Multiply the following 2x2 matrices:
A =
[1, -2]
[3, -5]
B =
[-1, -3]
[2, -5] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (1*-1) + (-2*2) = -5
C12 = (A11 * B12) + (A12 * B22) = (1*-3) + (-2*-5) = 7
C21 = (A21 * B11) + (A22 * B21) = (3*-1) + (-5*2) = -13
C22 = (A21 * B12) + (A22 * B22) = (3*-3) + (-5*-5) = 16
Res... |
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 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 quotient and remainder of 17387 divided by 10. Show the long division steps. | To divide 17387 by 10:
1. Determine how many times 10 goes into the leading digits of 17387.
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: 17387 / 10 = 1738 with a remainder of 7.
Proof: (1738 * 10)... |
Add the fractions 8/15 and 5/2. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 15 and 2, which is 30.
2. Convert 8/15 by multiplying top and bottom by 2: 16/30.
3. Convert 5/2 by multiplying top and bottom by 15: 75/30.
4. Add the numerators: 16 + 75 = 91.
5. The resulting fraction is 91/30.
Final Answer: 9... |
Find the roots of the quadratic equation: 6x^2 + 3x + -9 = 0 | 1. Identify coefficients: a=6, b=3, c=-9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 9 - -216 = 225.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-3 ± √225) / 12.
5. Calculate roots: x1 = 1.0, x2 = -1.5.
Final Answer: x = 1.0, -1.5 |
Compute the quotient and remainder of 10440 divided by 95. Show the long division steps. | To divide 10440 by 95:
1. Determine how many times 95 goes into the leading digits of 10440.
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: 10440 / 95 = 109 with a remainder of 85.
Proof: (109 * 95) ... |
Evaluate tan(30°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(30°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.5774 (exact form depends on root simplification if applicable) |
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) |
Multiply the following 2x2 matrices:
A =
[-2, -3]
[-2, 5]
B =
[-1, -1]
[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) = (-2*-1) + (-3*1) = -1
C12 = (A11 * B12) + (A12 * B22) = (-2*-1) + (-3*-5) = 17
C21 = (A21 * B11) + (A22 * B21) = (-2*-1) + (5*1) = 7
C22 = (A21 * B12) + (A22 * B22) = (-2*-1) + (5*-5) = -23
R... |
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 |
Multiply the following 2x2 matrices:
A =
[-3, 1]
[-5, -5]
B =
[-3, -1]
[4, -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) + (1*4) = 13
C12 = (A11 * B12) + (A12 * B22) = (-3*-1) + (1*-2) = 1
C21 = (A21 * B11) + (A22 * B21) = (-5*-3) + (-5*4) = -5
C22 = (A21 * B12) + (A22 * B22) = (-5*-1) + (-5*-2) = 15
Re... |
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 |
Calculate the mean, population variance, and standard deviation for the dataset: [16, 9, 12, 11, 15] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (16 + 9 + 12 + 11 + 15) / 5 = 63 / 5 = 12.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [3.4, -3.6, -0.6, -1.6, 2.4]
Squared Differences: [11.56, 12.96, 0.36, 2.56, 5.76]
... |
Multiply the following 2x2 matrices:
A =
[0, 3]
[-2, -3]
B =
[3, 1]
[5, -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) + (3*5) = 15
C12 = (A11 * B12) + (A12 * B22) = (0*1) + (3*-5) = -15
C21 = (A21 * B11) + (A22 * B21) = (-2*3) + (-3*5) = -21
C22 = (A21 * B12) + (A22 * B22) = (-2*1) + (-3*-5) = 13
Resul... |
Compute the quotient and remainder of 12190 divided by 51. Show the long division steps. | To divide 12190 by 51:
1. Determine how many times 51 goes into the leading digits of 12190.
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: 12190 / 51 = 239 with a remainder of 1.
Proof: (239 * 51) +... |
Calculate the mean, population variance, and standard deviation for the dataset: [1, 2, 10, 9, 6] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (1 + 2 + 10 + 9 + 6) / 5 = 28 / 5 = 5.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-4.6, -3.6, 4.4, 3.4, 0.4]
Squared Differences: [21.16, 12.96, 19.36, 11.56, 0.16]
... |
Add the fractions 2/10 and 7/7. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 10 and 7, which is 70.
2. Convert 2/10 by multiplying top and bottom by 7: 14/70.
3. Convert 7/7 by multiplying top and bottom by 10: 70/70.
4. Add the numerators: 14 + 70 = 84.
5. The resulting fraction is 84/70.
6. Simplify by ... |
Find the roots of the quadratic equation: 8x^2 + 0x + 4 = 0 | 1. Identify coefficients: a=8, b=0, c=4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 0^2 - 4(8)(4) = 0 - 128 = -128.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (0 ± √128i) / 16.
Final Answer: x = 0.0 ± 0.7071067811865476i |
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 |
Add the fractions 13/7 and 7/11. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 7 and 11, which is 77.
2. Convert 13/7 by multiplying top and bottom by 11: 143/77.
3. Convert 7/11 by multiplying top and bottom by 7: 49/77.
4. Add the numerators: 143 + 49 = 192.
5. The resulting fraction is 192/77.
Final Answ... |
Add the fractions 5/8 and 15/3. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 8 and 3, which is 24.
2. Convert 5/8 by multiplying top and bottom by 3: 15/24.
3. Convert 15/3 by multiplying top and bottom by 8: 120/24.
4. Add the numerators: 15 + 120 = 135.
5. The resulting fraction is 135/24.
6. Simplify b... |
Find the roots of the quadratic equation: 8x^2 + 5x + -3 = 0 | 1. Identify coefficients: a=8, b=5, c=-3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 25 - -96 = 121.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-5 ± √121) / 16.
5. Calculate roots: x1 = 0.375, x2 = -1.0.
Final Answer: x = 0.375, -1.0 |
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 |
Multiply the following 2x2 matrices:
A =
[3, -3]
[1, -3]
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) = (3*-1) + (-3*-2) = 3
C12 = (A11 * B12) + (A12 * B22) = (3*4) + (-3*1) = 9
C21 = (A21 * B11) + (A22 * B21) = (1*-1) + (-3*-2) = 5
C22 = (A21 * B12) + (A22 * B22) = (1*4) + (-3*1) = 1
Result Ma... |
Add the fractions 13/13 and 12/15. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 13 and 15, which is 195.
2. Convert 13/13 by multiplying top and bottom by 15: 195/195.
3. Convert 12/15 by multiplying top and bottom by 13: 156/195.
4. Add the numerators: 195 + 156 = 351.
5. The resulting fraction is 351/195.
... |
In a right-angled triangle, the lengths of the two legs are a = 16 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: 16^2 + 15^2 = c^2
2. Calculate the squares: 256 + 225 = c^2
3. Sum the squares: 481 = c^2
4. Take the square root of both sides: c = √481
Final A... |
In a right-angled triangle, the lengths of the two legs are a = 15 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: 15^2 + 4^2 = c^2
2. Calculate the squares: 225 + 16 = c^2
3. Sum the squares: 241 = c^2
4. Take the square root of both sides: c = √241
Final Ans... |
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 ... |
Compute the quotient and remainder of 41412 divided by 99. Show the long division steps. | To divide 41412 by 99:
1. Determine how many times 99 goes into the leading digits of 41412.
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: 41412 / 99 = 418 with a remainder of 30.
Proof: (418 * 99) ... |
Calculate the mean, population variance, and standard deviation for the dataset: [16, 8, 19, 9, 5] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (16 + 8 + 19 + 9 + 5) / 5 = 57 / 5 = 11.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [4.6, -3.4, 7.6, -2.4, -6.4]
Squared Differences: [21.16, 11.56, 57.76, 5.76, 40.96]
... |
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 ... |
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 |
In a right-angled triangle, the lengths of the two legs are a = 14 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: 14^2 + 3^2 = c^2
2. Calculate the squares: 196 + 9 = c^2
3. Sum the squares: 205 = c^2
4. Take the square root of both sides: c = √205
Final Answ... |
Find the roots of the quadratic equation: 10x^2 + -9x + 8 = 0 | 1. Identify coefficients: a=10, b=-9, c=8.
2. Calculate the discriminant: Δ = b^2 - 4ac = -9^2 - 4(10)(8) = 81 - 320 = -239.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (9 ± √239i) / 20.
Final Answer: x = 0.45 ± 0.7729812416870153i |
In a right-angled triangle, the lengths of the two legs are a = 16 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: 16^2 + 4^2 = c^2
2. Calculate the squares: 256 + 16 = c^2
3. Sum the squares: 272 = c^2
4. Take the square root of both sides: c = √272
Final Ans... |
Compute the quotient and remainder of 74951 divided by 94. Show the long division steps. | To divide 74951 by 94:
1. Determine how many times 94 goes into the leading digits of 74951.
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: 74951 / 94 = 797 with a remainder of 33.
Proof: (797 * 94) ... |
Compute the quotient and remainder of 44197 divided by 86. Show the long division steps. | To divide 44197 by 86:
1. Determine how many times 86 goes into the leading digits of 44197.
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: 44197 / 86 = 513 with a remainder of 79.
Proof: (513 * 86) ... |
Compute the quotient and remainder of 52920 divided by 64. Show the long division steps. | To divide 52920 by 64:
1. Determine how many times 64 goes into the leading digits of 52920.
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: 52920 / 64 = 826 with a remainder of 56.
Proof: (826 * 64) ... |
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 +... |
Compute the quotient and remainder of 1065 divided by 16. Show the long division steps. | To divide 1065 by 16:
1. Determine how many times 16 goes into the leading digits of 1065.
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: 1065 / 16 = 66 with a remainder of 9.
Proof: (66 * 16) + 9 = ... |
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