prompt stringlengths 32 115 | completion stringlengths 167 566 |
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Multiply the following 2x2 matrices:
A =
[5, -2]
[1, 0]
B =
[2, 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) = (5*2) + (-2*2) = 6
C12 = (A11 * B12) + (A12 * B22) = (5*5) + (-2*-2) = 29
C21 = (A21 * B11) + (A22 * B21) = (1*2) + (0*2) = 2
C22 = (A21 * B12) + (A22 * B22) = (1*5) + (0*-2) = 5
Result Matri... |
Add the fractions 9/4 and 7/8. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 4 and 8, which is 8.
2. Convert 9/4 by multiplying top and bottom by 2: 18/8.
3. Convert 7/8 by multiplying top and bottom by 1: 7/8.
4. Add the numerators: 18 + 7 = 25.
5. The resulting fraction is 25/8.
Final Answer: 25/8 |
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 |
In a right-angled triangle, the lengths of the two legs are a = 18 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: 18^2 + 18^2 = c^2
2. Calculate the squares: 324 + 324 = c^2
3. Sum the squares: 648 = c^2
4. Take the square root of both sides: c = √648
Final A... |
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) |
Compute the quotient and remainder of 57617 divided by 56. Show the long division steps. | To divide 57617 by 56:
1. Determine how many times 56 goes into the leading digits of 57617.
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: 57617 / 56 = 1028 with a remainder of 49.
Proof: (1028 * 56... |
Compute the quotient and remainder of 55747 divided by 9. Show the long division steps. | To divide 55747 by 9:
1. Determine how many times 9 goes into the leading digits of 55747.
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: 55747 / 9 = 6194 with a remainder of 1.
Proof: (6194 * 9) + 1... |
Compute the quotient and remainder of 60395 divided by 74. Show the long division steps. | To divide 60395 by 74:
1. Determine how many times 74 goes into the leading digits of 60395.
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: 60395 / 74 = 816 with a remainder of 11.
Proof: (816 * 74) ... |
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 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 |
In a right-angled triangle, the lengths of the two legs are a = 14 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: 14^2 + 15^2 = c^2
2. Calculate the squares: 196 + 225 = c^2
3. Sum the squares: 421 = c^2
4. Take the square root of both sides: c = √421
Final A... |
Compute the indefinite integral of f(x) = 2x^2. | To find the indefinite integral ∫ 2x^2 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 2 + 1 = 3.
2. Divide the coefficient by the new exponent: 2 / 3 = 2/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 2x^2 dx = 2/3x^3 + C |
Add the fractions 13/2 and 8/14. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 2 and 14, which is 14.
2. Convert 13/2 by multiplying top and bottom by 7: 91/14.
3. Convert 8/14 by multiplying top and bottom by 1: 8/14.
4. Add the numerators: 91 + 8 = 99.
5. The resulting fraction is 99/14.
Final Answer: 99/... |
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 ... |
Multiply the following 2x2 matrices:
A =
[1, 0]
[-1, -4]
B =
[4, -3]
[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) = (1*4) + (0*3) = 4
C12 = (A11 * B12) + (A12 * B22) = (1*-3) + (0*4) = -3
C21 = (A21 * B11) + (A22 * B21) = (-1*4) + (-4*3) = -16
C22 = (A21 * B12) + (A22 * B22) = (-1*-3) + (-4*4) = -13
Result... |
Find the roots of the quadratic equation: 2x^2 + -7x + 0 = 0 | 1. Identify coefficients: a=2, 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) / 4.
5. Calculate roots: x1 = 3.5, x2 = 0.0.
Final Answer: x = 3.5, 0.0 |
Compute the quotient and remainder of 47052 divided by 27. Show the long division steps. | To divide 47052 by 27:
1. Determine how many times 27 goes into the leading digits of 47052.
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: 47052 / 27 = 1742 with a remainder of 18.
Proof: (1742 * 27... |
Compute the quotient and remainder of 1626 divided by 45. Show the long division steps. | To divide 1626 by 45:
1. Determine how many times 45 goes into the leading digits of 1626.
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: 1626 / 45 = 36 with a remainder of 6.
Proof: (36 * 45) + 6 = ... |
Find the roots of the quadratic equation: 10x^2 + 1x + -7 = 0 | 1. Identify coefficients: a=10, b=1, c=-7.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1 - -280 = 281.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-1 ± √281) / 20.
5. Calculate roots: x1 = 0.7881527307120105, x2 = -0.8881527307120105.
Final... |
Add the fractions 1/7 and 2/9. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 7 and 9, which is 63.
2. Convert 1/7 by multiplying top and bottom by 9: 9/63.
3. Convert 2/9 by multiplying top and bottom by 7: 14/63.
4. Add the numerators: 9 + 14 = 23.
5. The resulting fraction is 23/63.
Final Answer: 23/63 |
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 91780 divided by 69. Show the long division steps. | To divide 91780 by 69:
1. Determine how many times 69 goes into the leading digits of 91780.
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: 91780 / 69 = 1330 with a remainder of 10.
Proof: (1330 * 69... |
Multiply the following 2x2 matrices:
A =
[2, 3]
[-5, 3]
B =
[-3, 2]
[2, 2] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (2*-3) + (3*2) = 0
C12 = (A11 * B12) + (A12 * B22) = (2*2) + (3*2) = 10
C21 = (A21 * B11) + (A22 * B21) = (-5*-3) + (3*2) = 21
C22 = (A21 * B12) + (A22 * B22) = (-5*2) + (3*2) = -4
Result Mat... |
Calculate the mean, population variance, and standard deviation for the dataset: [8, 5, 11, 16, 15] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (8 + 5 + 11 + 16 + 15) / 5 = 55 / 5 = 11.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-3.0, -6.0, 0.0, 5.0, 4.0]
Squared Differences: [9.0, 36.0, 0.0, 25.0, 16.0]
Sum... |
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) |
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 |
Compute the indefinite integral of f(x) = 4x^4. | To find the indefinite integral ∫ 4x^4 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 4 + 1 = 5.
2. Divide the coefficient by the new exponent: 4 / 5 = 4/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 4x^4 dx = 4/5x^5 + C |
Calculate the mean, population variance, and standard deviation for the dataset: [13, 15, 7, 20, 19] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (13 + 15 + 7 + 20 + 19) / 5 = 74 / 5 = 14.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-1.8, 0.2, -7.8, 5.2, 4.2]
Squared Differences: [3.24, 0.04, 60.84, 27.04, 17.64]
... |
Compute the quotient and remainder of 72499 divided by 56. Show the long division steps. | To divide 72499 by 56:
1. Determine how many times 56 goes into the leading digits of 72499.
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: 72499 / 56 = 1294 with a remainder of 35.
Proof: (1294 * 56... |
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 |
In a right-angled triangle, the lengths of the two legs are a = 4 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: 4^2 + 19^2 = c^2
2. Calculate the squares: 16 + 361 = c^2
3. Sum the squares: 377 = c^2
4. Take the square root of both sides: c = √377
Final Ans... |
Calculate the mean, population variance, and standard deviation for the dataset: [14, 14, 6, 17, 14] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (14 + 14 + 6 + 17 + 14) / 5 = 65 / 5 = 13.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [1.0, 1.0, -7.0, 4.0, 1.0]
Squared Differences: [1.0, 1.0, 49.0, 16.0, 1.0]
Sum ... |
In a right-angled triangle, the lengths of the two legs are a = 19 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: 19^2 + 19^2 = c^2
2. Calculate the squares: 361 + 361 = c^2
3. Sum the squares: 722 = c^2
4. Take the square root of both sides: c = √722
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 ... |
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) |
Evaluate the logarithm: log_5(3125) | To evaluate log_5(3125), we ask the question: '5 raised to what power equals 3125?'
Let x be the unknown power: 5^x = 3125
Since 5^5 = 3125, it follows that x = 5.
Final Answer: 5 |
Evaluate 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) |
Multiply the following 2x2 matrices:
A =
[-2, 3]
[4, -3]
B =
[-3, -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) = (-2*-3) + (3*1) = 9
C12 = (A11 * B12) + (A12 * B22) = (-2*-2) + (3*1) = 7
C21 = (A21 * B11) + (A22 * B21) = (4*-3) + (-3*1) = -15
C22 = (A21 * B12) + (A22 * B22) = (4*-2) + (-3*1) = -11
Resul... |
Compute the quotient and remainder of 60878 divided by 98. Show the long division steps. | To divide 60878 by 98:
1. Determine how many times 98 goes into the leading digits of 60878.
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: 60878 / 98 = 621 with a remainder of 20.
Proof: (621 * 98) ... |
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 71539 divided by 71. Show the long division steps. | To divide 71539 by 71:
1. Determine how many times 71 goes into the leading digits of 71539.
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: 71539 / 71 = 1007 with a remainder of 42.
Proof: (1007 * 71... |
In a right-angled triangle, the lengths of the two legs are a = 14 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: 14^2 + 19^2 = c^2
2. Calculate the squares: 196 + 361 = c^2
3. Sum the squares: 557 = c^2
4. Take the square root of both sides: c = √557
Final A... |
Find the roots of the quadratic equation: 5x^2 + 8x + 4 = 0 | 1. Identify coefficients: a=5, b=8, c=4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 8^2 - 4(5)(4) = 64 - 80 = -16.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-8 ± √16i) / 10.
Final Answer: x = -0.8 ± 0.4i |
Compute the quotient and remainder of 57226 divided by 82. Show the long division steps. | To divide 57226 by 82:
1. Determine how many times 82 goes into the leading digits of 57226.
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: 57226 / 82 = 697 with a remainder of 72.
Proof: (697 * 82) ... |
Multiply the following 2x2 matrices:
A =
[-4, 0]
[4, -5]
B =
[1, 3]
[2, 0] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-4*1) + (0*2) = -4
C12 = (A11 * B12) + (A12 * B22) = (-4*3) + (0*0) = -12
C21 = (A21 * B11) + (A22 * B21) = (4*1) + (-5*2) = -6
C22 = (A21 * B12) + (A22 * B22) = (4*3) + (-5*0) = 12
Result M... |
Multiply the following 2x2 matrices:
A =
[5, 1]
[1, -1]
B =
[-5, -4]
[-3, -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) = (5*-5) + (1*-3) = -28
C12 = (A11 * B12) + (A12 * B22) = (5*-4) + (1*-2) = -22
C21 = (A21 * B11) + (A22 * B21) = (1*-5) + (-1*-3) = -2
C22 = (A21 * B12) + (A22 * B22) = (1*-4) + (-1*-2) = -2
R... |
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 ... |
Calculate the mean, population variance, and standard deviation for the dataset: [9, 11, 14, 9, 3] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (9 + 11 + 14 + 9 + 3) / 5 = 46 / 5 = 9.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-0.2, 1.8, 4.8, -0.2, -6.2]
Squared Differences: [0.04, 3.24, 23.04, 0.04, 38.44]
... |
Add the fractions 1/6 and 2/2. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 6 and 2, which is 6.
2. Convert 1/6 by multiplying top and bottom by 1: 1/6.
3. Convert 2/2 by multiplying top and bottom by 3: 6/6.
4. Add the numerators: 1 + 6 = 7.
5. The resulting fraction is 7/6.
Final Answer: 7/6 |
Compute the indefinite integral of f(x) = 7x^3. | To find the indefinite integral ∫ 7x^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: 7 / 4 = 7/4.
3. Combine and add the constant of integration C.
Final Answer: ∫ 7x^3 dx = 7/4x^4 + C |
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) |
Multiply the following 2x2 matrices:
A =
[3, -5]
[-4, 3]
B =
[-3, -2]
[-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) = (3*-3) + (-5*-1) = -4
C12 = (A11 * B12) + (A12 * B22) = (3*-2) + (-5*-5) = 19
C21 = (A21 * B11) + (A22 * B21) = (-4*-3) + (3*-1) = 9
C22 = (A21 * B12) + (A22 * B22) = (-4*-2) + (3*-5) = -7
Re... |
Multiply the following 2x2 matrices:
A =
[2, 0]
[3, -5]
B =
[-3, -3]
[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) = (2*-3) + (0*5) = -6
C12 = (A11 * B12) + (A12 * B22) = (2*-3) + (0*4) = -6
C21 = (A21 * B11) + (A22 * B21) = (3*-3) + (-5*5) = -34
C22 = (A21 * B12) + (A22 * B22) = (3*-3) + (-5*4) = -29
Resul... |
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 |
Add the fractions 7/12 and 8/5. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 12 and 5, which is 60.
2. Convert 7/12 by multiplying top and bottom by 5: 35/60.
3. Convert 8/5 by multiplying top and bottom by 12: 96/60.
4. Add the numerators: 35 + 96 = 131.
5. The resulting fraction is 131/60.
Final Answer:... |
Compute the quotient and remainder of 9684 divided by 50. Show the long division steps. | To divide 9684 by 50:
1. Determine how many times 50 goes into the leading digits of 9684.
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: 9684 / 50 = 193 with a remainder of 34.
Proof: (193 * 50) + 3... |
Find the roots of the quadratic equation: 2x^2 + -10x + 0 = 0 | 1. Identify coefficients: a=2, b=-10, c=0.
2. Calculate the discriminant: Δ = b^2 - 4ac = 100 - 0 = 100.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (10 ± √100) / 4.
5. Calculate roots: x1 = 5.0, x2 = 0.0.
Final Answer: x = 5.0, 0.0 |
In a right-angled triangle, the lengths of the two legs are a = 17 and b = 13. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 17^2 + 13^2 = c^2
2. Calculate the squares: 289 + 169 = c^2
3. Sum the squares: 458 = c^2
4. Take the square root of both sides: c = √458
Final A... |
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 |
Find the derivative of f(x) = (4x + 2)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 4x + 2.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 2. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 2)^2 * 4 = 12(4x ... |
Find the roots of the quadratic equation: 10x^2 + -10x + 7 = 0 | 1. Identify coefficients: a=10, b=-10, c=7.
2. Calculate the discriminant: Δ = b^2 - 4ac = -10^2 - 4(10)(7) = 100 - 280 = -180.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (10 ± √180i) / 20.
Final Answer: x = 0.5 ± 0.6708203932499369i |
Multiply the following 2x2 matrices:
A =
[1, 4]
[-1, -4]
B =
[5, 4]
[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) = (1*5) + (4*1) = 9
C12 = (A11 * B12) + (A12 * B22) = (1*4) + (4*-2) = -4
C21 = (A21 * B11) + (A22 * B21) = (-1*5) + (-4*1) = -9
C22 = (A21 * B12) + (A22 * B22) = (-1*4) + (-4*-2) = 4
Result Ma... |
Add the fractions 14/6 and 7/11. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 6 and 11, which is 66.
2. Convert 14/6 by multiplying top and bottom by 11: 154/66.
3. Convert 7/11 by multiplying top and bottom by 6: 42/66.
4. Add the numerators: 154 + 42 = 196.
5. The resulting fraction is 196/66.
6. Simplif... |
Evaluate the logarithm: log_4(256) | To evaluate log_4(256), we ask the question: '4 raised to what power equals 256?'
Let x be the unknown power: 4^x = 256
Since 4^4 = 256, it follows that x = 4.
Final Answer: 4 |
Compute the quotient and remainder of 79946 divided by 45. Show the long division steps. | To divide 79946 by 45:
1. Determine how many times 45 goes into the leading digits of 79946.
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: 79946 / 45 = 1776 with a remainder of 26.
Proof: (1776 * 45... |
Find the roots of the quadratic equation: 8x^2 + 10x + 0 = 0 | 1. Identify coefficients: a=8, b=10, c=0.
2. Calculate the discriminant: Δ = b^2 - 4ac = 100 - 0 = 100.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-10 ± √100) / 16.
5. Calculate roots: x1 = 0.0, x2 = -1.25.
Final Answer: x = 0.0, -1.25 |
Compute the indefinite integral of f(x) = 9x^4. | To find the indefinite integral ∫ 9x^4 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 4 + 1 = 5.
2. Divide the coefficient by the new exponent: 9 / 5 = 9/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 9x^4 dx = 9/5x^5 + C |
Multiply the following 2x2 matrices:
A =
[0, -1]
[-5, 1]
B =
[-5, 4]
[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) = (0*-5) + (-1*3) = -3
C12 = (A11 * B12) + (A12 * B22) = (0*4) + (-1*1) = -1
C21 = (A21 * B11) + (A22 * B21) = (-5*-5) + (1*3) = 28
C22 = (A21 * B12) + (A22 * B22) = (-5*4) + (1*1) = -19
Result... |
Multiply the following 2x2 matrices:
A =
[-4, -1]
[3, -2]
B =
[4, -3]
[-1, 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) = (-4*4) + (-1*-1) = -15
C12 = (A11 * B12) + (A12 * B22) = (-4*-3) + (-1*0) = 12
C21 = (A21 * B11) + (A22 * B21) = (3*4) + (-2*-1) = 14
C22 = (A21 * B12) + (A22 * B22) = (3*-3) + (-2*0) = -9
Re... |
Add the fractions 4/3 and 6/11. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 3 and 11, which is 33.
2. Convert 4/3 by multiplying top and bottom by 11: 44/33.
3. Convert 6/11 by multiplying top and bottom by 3: 18/33.
4. Add the numerators: 44 + 18 = 62.
5. The resulting fraction is 62/33.
Final Answer: 6... |
Evaluate the logarithm: log_2(8) | To evaluate log_2(8), we ask the question: '2 raised to what power equals 8?'
Let x be the unknown power: 2^x = 8
Since 2^3 = 8, it follows that x = 3.
Final Answer: 3 |
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) |
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) = 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 |
Multiply the following 2x2 matrices:
A =
[4, -4]
[2, 2]
B =
[4, 1]
[4, -3] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (4*4) + (-4*4) = 0
C12 = (A11 * B12) + (A12 * B22) = (4*1) + (-4*-3) = 16
C21 = (A21 * B11) + (A22 * B21) = (2*4) + (2*4) = 16
C22 = (A21 * B12) + (A22 * B22) = (2*1) + (2*-3) = -4
Result Mat... |
Multiply the following 2x2 matrices:
A =
[-4, -5]
[-3, -1]
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) = (-4*-5) + (-5*-5) = 45
C12 = (A11 * B12) + (A12 * B22) = (-4*-2) + (-5*-5) = 33
C21 = (A21 * B11) + (A22 * B21) = (-3*-5) + (-1*-5) = 20
C22 = (A21 * B12) + (A22 * B22) = (-3*-2) + (-1*-5) = 1... |
Multiply the following 2x2 matrices:
A =
[-3, 1]
[-2, 1]
B =
[3, -5]
[2, -4] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-3*3) + (1*2) = -7
C12 = (A11 * B12) + (A12 * B22) = (-3*-5) + (1*-4) = 11
C21 = (A21 * B11) + (A22 * B21) = (-2*3) + (1*2) = -4
C22 = (A21 * B12) + (A22 * B22) = (-2*-5) + (1*-4) = 6
Result... |
In a right-angled triangle, the lengths of the two legs are a = 7 and b = 5. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 7^2 + 5^2 = c^2
2. Calculate the squares: 49 + 25 = c^2
3. Sum the squares: 74 = c^2
4. Take the square root of both sides: c = √74
Final Answer:... |
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 tan(90°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(90°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 16331239353195370.0000 (exact form depends on root simplification if applicable) |
Compute the quotient and remainder of 8108 divided by 25. Show the long division steps. | To divide 8108 by 25:
1. Determine how many times 25 goes into the leading digits of 8108.
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: 8108 / 25 = 324 with a remainder of 8.
Proof: (324 * 25) + 8 ... |
Compute the indefinite integral of f(x) = 1x^5. | To find the indefinite integral ∫ 1x^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: 1 / 6 = 1/6.
3. Combine and add the constant of integration C.
Final Answer: ∫ 1x^5 dx = 1/6x^6 + C |
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, 9, 7, 4, 4] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (15 + 9 + 7 + 4 + 4) / 5 = 39 / 5 = 7.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [7.2, 1.2, -0.8, -3.8, -3.8]
Squared Differences: [51.84, 1.44, 0.64, 14.44, 14.44]
... |
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 quotient and remainder of 16568 divided by 78. Show the long division steps. | To divide 16568 by 78:
1. Determine how many times 78 goes into the leading digits of 16568.
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: 16568 / 78 = 212 with a remainder of 32.
Proof: (212 * 78) ... |
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 ... |
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) |
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) |
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 |
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 ... |
Calculate the mean, population variance, and standard deviation for the dataset: [8, 7, 11, 13, 15] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (8 + 7 + 11 + 13 + 15) / 5 = 54 / 5 = 10.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-2.8, -3.8, 0.2, 2.2, 4.2]
Squared Differences: [7.84, 14.44, 0.04, 4.84, 17.64]
... |
Compute the quotient and remainder of 64409 divided by 69. Show the long division steps. | To divide 64409 by 69:
1. Determine how many times 69 goes into the leading digits of 64409.
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: 64409 / 69 = 933 with a remainder of 32.
Proof: (933 * 69) ... |
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 |
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... |
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 +... |
In a right-angled triangle, the lengths of the two legs are a = 7 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: 7^2 + 8^2 = c^2
2. Calculate the squares: 49 + 64 = c^2
3. Sum the squares: 113 = c^2
4. Take the square root of both sides: c = √113
Final Answe... |
Calculate the mean, population variance, and standard deviation for the dataset: [11, 3, 8, 9, 20] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (11 + 3 + 8 + 9 + 20) / 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, -7.2, -2.2, -1.2, 9.8]
Squared Differences: [0.64, 51.84, 4.84, 1.44, 96.04]
... |
Find the roots of the quadratic equation: 3x^2 + 7x + 7 = 0 | 1. Identify coefficients: a=3, b=7, c=7.
2. Calculate the discriminant: Δ = b^2 - 4ac = 7^2 - 4(3)(7) = 49 - 84 = -35.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-7 ± √35i) / 6.
Final Answer: x = -1.1666666666666667 ± 0.9860132971832694i |
Calculate the mean, population variance, and standard deviation for the dataset: [10, 7, 6, 5, 17] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (10 + 7 + 6 + 5 + 17) / 5 = 45 / 5 = 9.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [1.0, -2.0, -3.0, -4.0, 8.0]
Squared Differences: [1.0, 4.0, 9.0, 16.0, 64.0]
Sum o... |
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