prompt stringlengths 32 115 | completion stringlengths 167 566 |
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Multiply the following 2x2 matrices:
A =
[5, -1]
[1, -1]
B =
[2, 4]
[-2, 3] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (5*2) + (-1*-2) = 12
C12 = (A11 * B12) + (A12 * B22) = (5*4) + (-1*3) = 17
C21 = (A21 * B11) + (A22 * B21) = (1*2) + (-1*-2) = 4
C22 = (A21 * B12) + (A22 * B22) = (1*4) + (-1*3) = 1
Result Ma... |
Compute the indefinite integral of f(x) = 9x^5. | To find the indefinite integral ∫ 9x^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: 9 / 6 = 3/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 9x^5 dx = 3/2x^6 + C |
Multiply the following 2x2 matrices:
A =
[-2, 2]
[3, -3]
B =
[4, 2]
[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) = (-2*4) + (2*3) = -2
C12 = (A11 * B12) + (A12 * B22) = (-2*2) + (2*-2) = -8
C21 = (A21 * B11) + (A22 * B21) = (3*4) + (-3*3) = 3
C22 = (A21 * B12) + (A22 * B22) = (3*2) + (-3*-2) = 12
Result M... |
In a right-angled triangle, the lengths of the two legs are a = 14 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: 14^2 + 8^2 = c^2
2. Calculate the squares: 196 + 64 = c^2
3. Sum the squares: 260 = c^2
4. Take the square root of both sides: c = √260
Final Ans... |
In a right-angled triangle, the lengths of the two legs are a = 8 and b = 9. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 8^2 + 9^2 = c^2
2. Calculate the squares: 64 + 81 = c^2
3. Sum the squares: 145 = c^2
4. Take the square root of both sides: c = √145
Final Answe... |
Add the fractions 4/14 and 8/15. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 14 and 15, which is 210.
2. Convert 4/14 by multiplying top and bottom by 15: 60/210.
3. Convert 8/15 by multiplying top and bottom by 14: 112/210.
4. Add the numerators: 60 + 112 = 172.
5. The resulting fraction is 172/210.
6. S... |
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 |
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) |
Multiply the following 2x2 matrices:
A =
[1, -3]
[-2, -4]
B =
[-3, 3]
[0, -2] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (1*-3) + (-3*0) = -3
C12 = (A11 * B12) + (A12 * B22) = (1*3) + (-3*-2) = 9
C21 = (A21 * B11) + (A22 * B21) = (-2*-3) + (-4*0) = 6
C22 = (A21 * B12) + (A22 * B22) = (-2*3) + (-4*-2) = 2
Result... |
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 |
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 ... |
In a right-angled triangle, the lengths of the two legs are a = 19 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: 19^2 + 18^2 = c^2
2. Calculate the squares: 361 + 324 = c^2
3. Sum the squares: 685 = c^2
4. Take the square root of both sides: c = √685
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 +... |
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 ... |
Add the fractions 2/12 and 13/10. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 12 and 10, which is 60.
2. Convert 2/12 by multiplying top and bottom by 5: 10/60.
3. Convert 13/10 by multiplying top and bottom by 6: 78/60.
4. Add the numerators: 10 + 78 = 88.
5. The resulting fraction is 88/60.
6. Simplify b... |
Compute the indefinite integral of f(x) = 4x^1. | To find the indefinite integral ∫ 4x^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: 4 / 2 = 2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 4x^1 dx = 2x^2 + C |
Evaluate tan(90°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(90°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 16331239353195370.0000 (exact form depends on root simplification if applicable) |
Multiply the following 2x2 matrices:
A =
[2, 2]
[-4, 4]
B =
[2, -3]
[-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) = (2*2) + (2*-3) = -2
C12 = (A11 * B12) + (A12 * B22) = (2*-3) + (2*-3) = -12
C21 = (A21 * B11) + (A22 * B21) = (-4*2) + (4*-3) = -20
C22 = (A21 * B12) + (A22 * B22) = (-4*-3) + (4*-3) = 0
Resu... |
Calculate the mean, population variance, and standard deviation for the dataset: [9, 10, 14, 4, 19] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (9 + 10 + 14 + 4 + 19) / 5 = 56 / 5 = 11.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-2.2, -1.2, 2.8, -7.2, 7.8]
Squared Differences: [4.84, 1.44, 7.84, 51.84, 60.84]
... |
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) = 9x^2. | To find the indefinite integral ∫ 9x^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: 9 / 3 = 3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 9x^2 dx = 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 +... |
Compute the quotient and remainder of 55855 divided by 49. Show the long division steps. | To divide 55855 by 49:
1. Determine how many times 49 goes into the leading digits of 55855.
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: 55855 / 49 = 1139 with a remainder of 44.
Proof: (1139 * 49... |
In a right-angled triangle, the lengths of the two legs are a = 3 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: 3^2 + 19^2 = c^2
2. Calculate the squares: 9 + 361 = c^2
3. Sum the squares: 370 = c^2
4. Take the square root of both sides: c = √370
Final Answ... |
Find the roots of the quadratic equation: 9x^2 + -1x + 3 = 0 | 1. Identify coefficients: a=9, b=-1, c=3.
2. Calculate the discriminant: Δ = b^2 - 4ac = -1^2 - 4(9)(3) = 1 - 108 = -107.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (1 ± √107i) / 18.
Final Answer: x = 0.05555555555555555 ± 0.5746711351549223i |
Compute the quotient and remainder of 14390 divided by 69. Show the long division steps. | To divide 14390 by 69:
1. Determine how many times 69 goes into the leading digits of 14390.
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: 14390 / 69 = 208 with a remainder of 38.
Proof: (208 * 69) ... |
Compute the quotient and remainder of 6460 divided by 21. Show the long division steps. | To divide 6460 by 21:
1. Determine how many times 21 goes into the leading digits of 6460.
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: 6460 / 21 = 307 with a remainder of 13.
Proof: (307 * 21) + 1... |
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) |
Find the roots of the quadratic equation: 1x^2 + 4x + 4 = 0 | 1. Identify coefficients: a=1, b=4, c=4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 16 - 16 = 0.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-4 ± √0) / 2.
5. Calculate roots: x1 = -2.0, x2 = -2.0.
Final Answer: x = -2.0, -2.0 |
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 |
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 +... |
Multiply the following 2x2 matrices:
A =
[-3, -3]
[-5, -2]
B =
[1, -3]
[-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*1) + (-3*-4) = 9
C12 = (A11 * B12) + (A12 * B22) = (-3*-3) + (-3*-2) = 15
C21 = (A21 * B11) + (A22 * B21) = (-5*1) + (-2*-4) = 3
C22 = (A21 * B12) + (A22 * B22) = (-5*-3) + (-2*-2) = 19
R... |
Compute the quotient and remainder of 60108 divided by 25. Show the long division steps. | To divide 60108 by 25:
1. Determine how many times 25 goes into the leading digits of 60108.
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: 60108 / 25 = 2404 with a remainder of 8.
Proof: (2404 * 25)... |
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 ... |
Compute the indefinite integral of f(x) = 7x^2. | To find the indefinite integral ∫ 7x^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: 7 / 3 = 7/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 7x^2 dx = 7/3x^3 + C |
Compute the indefinite integral of f(x) = 7x^5. | To find the indefinite integral ∫ 7x^5 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 5 + 1 = 6.
2. Divide the coefficient by the new exponent: 7 / 6 = 7/6.
3. Combine and add the constant of integration C.
Final Answer: ∫ 7x^5 dx = 7/6x^6 + C |
Compute the indefinite integral of f(x) = 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 |
In a right-angled triangle, the lengths of the two legs are a = 7 and b = 7. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 7^2 + 7^2 = c^2
2. Calculate the squares: 49 + 49 = c^2
3. Sum the squares: 98 = c^2
4. Take the square root of both sides: c = √98
Final Answer:... |
Evaluate cos(60°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(60°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.5000 (exact form depends on root simplification if applicable) |
Compute the indefinite integral of f(x) = 5x^2. | To find the indefinite integral ∫ 5x^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: 5 / 3 = 5/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 5x^2 dx = 5/3x^3 + C |
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 |
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 |
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 roots of the quadratic equation: 1x^2 + -6x + 7 = 0 | 1. Identify coefficients: a=1, b=-6, c=7.
2. Calculate the discriminant: Δ = b^2 - 4ac = 36 - 28 = 8.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (6 ± √8) / 2.
5. Calculate roots: x1 = 4.414213562373095, x2 = 1.5857864376269049.
Final Answer: x... |
Find the roots of the quadratic equation: 9x^2 + 5x + -10 = 0 | 1. Identify coefficients: a=9, b=5, c=-10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 25 - -360 = 385.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-5 ± √385) / 18.
5. Calculate roots: x1 = 0.8123009372415879, x2 = -1.3678564927971435.
Fina... |
Add the fractions 10/4 and 15/6. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 4 and 6, which is 12.
2. Convert 10/4 by multiplying top and bottom by 3: 30/12.
3. Convert 15/6 by multiplying top and bottom by 2: 30/12.
4. Add the numerators: 30 + 30 = 60.
5. The resulting fraction is 60/12.
6. Simplify by d... |
Multiply the following 2x2 matrices:
A =
[-1, -5]
[-1, 4]
B =
[4, 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) = (-1*4) + (-5*-1) = 1
C12 = (A11 * B12) + (A12 * B22) = (-1*1) + (-5*-2) = 9
C21 = (A21 * B11) + (A22 * B21) = (-1*4) + (4*-1) = -8
C22 = (A21 * B12) + (A22 * B22) = (-1*1) + (4*-2) = -9
Resul... |
Add the fractions 11/2 and 5/4. 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 11/2 by multiplying top and bottom by 2: 22/4.
3. Convert 5/4 by multiplying top and bottom by 1: 5/4.
4. Add the numerators: 22 + 5 = 27.
5. The resulting fraction is 27/4.
Final Answer: 27/4 |
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 ... |
Multiply the following 2x2 matrices:
A =
[1, -3]
[-4, -4]
B =
[2, -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*2) + (-3*2) = -4
C12 = (A11 * B12) + (A12 * B22) = (1*-1) + (-3*5) = -16
C21 = (A21 * B11) + (A22 * B21) = (-4*2) + (-4*2) = -16
C22 = (A21 * B12) + (A22 * B22) = (-4*-1) + (-4*5) = -16
Re... |
Find the roots of the quadratic equation: 4x^2 + 4x + -10 = 0 | 1. Identify coefficients: a=4, b=4, c=-10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 16 - -160 = 176.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-4 ± √176) / 8.
5. Calculate roots: x1 = 1.1583123951777, x2 = -2.1583123951777.
Final Answe... |
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: [7, 14, 16, 17, 4] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (7 + 14 + 16 + 17 + 4) / 5 = 58 / 5 = 11.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-4.6, 2.4, 4.4, 5.4, -7.6]
Squared Differences: [21.16, 5.76, 19.36, 29.16, 57.76]
... |
Find the roots of the quadratic equation: 5x^2 + 7x + 4 = 0 | 1. Identify coefficients: a=5, b=7, c=4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 7^2 - 4(5)(4) = 49 - 80 = -31.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-7 ± √31i) / 10.
Final Answer: x = -0.7 ± 0.5567764362830021i |
Multiply the following 2x2 matrices:
A =
[2, 5]
[4, 5]
B =
[0, -4]
[-1, -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*0) + (5*-1) = -5
C12 = (A11 * B12) + (A12 * B22) = (2*-4) + (5*-4) = -28
C21 = (A21 * B11) + (A22 * B21) = (4*0) + (5*-1) = -5
C22 = (A21 * B12) + (A22 * B22) = (4*-4) + (5*-4) = -36
Resul... |
Evaluate tan(60°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(60°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 1.7321 (exact form depends on root simplification if applicable) |
Find the roots of the quadratic equation: 5x^2 + 3x + 5 = 0 | 1. Identify coefficients: a=5, b=3, c=5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 3^2 - 4(5)(5) = 9 - 100 = -91.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-3 ± √91i) / 10.
Final Answer: x = -0.3 ± 0.9539392014169457i |
In a right-angled triangle, the lengths of the two legs are a = 10 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: 10^2 + 15^2 = c^2
2. Calculate the squares: 100 + 225 = c^2
3. Sum the squares: 325 = c^2
4. Take the square root of both sides: c = √325
Final A... |
Compute the indefinite integral of f(x) = 7x^2. | To find the indefinite integral ∫ 7x^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: 7 / 3 = 7/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 7x^2 dx = 7/3x^3 + C |
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) |
Add the fractions 15/11 and 13/2. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 11 and 2, which is 22.
2. Convert 15/11 by multiplying top and bottom by 2: 30/22.
3. Convert 13/2 by multiplying top and bottom by 11: 143/22.
4. Add the numerators: 30 + 143 = 173.
5. The resulting fraction is 173/22.
Final Ans... |
Compute the quotient and remainder of 59730 divided by 20. Show the long division steps. | To divide 59730 by 20:
1. Determine how many times 20 goes into the leading digits of 59730.
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: 59730 / 20 = 2986 with a remainder of 10.
Proof: (2986 * 20... |
Find the roots of the quadratic equation: 6x^2 + 10x + 4 = 0 | 1. Identify coefficients: a=6, b=10, c=4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 100 - 96 = 4.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-10 ± √4) / 12.
5. Calculate roots: x1 = -0.6666666666666666, x2 = -1.0.
Final Answer: x = -0.66... |
Compute the indefinite integral of f(x) = 7x^1. | To find the indefinite integral ∫ 7x^1 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 1 + 1 = 2.
2. Divide the coefficient by the new exponent: 7 / 2 = 7/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 7x^1 dx = 7/2x^2 + C |
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) |
Find the roots of the quadratic equation: 5x^2 + 10x + -3 = 0 | 1. Identify coefficients: a=5, b=10, c=-3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 100 - -60 = 160.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-10 ± √160) / 10.
5. Calculate roots: x1 = 0.2649110640673518, x2 = -2.264911064067352.
Fina... |
Find the roots of the quadratic equation: 1x^2 + -3x + -8 = 0 | 1. Identify coefficients: a=1, b=-3, c=-8.
2. Calculate the discriminant: Δ = b^2 - 4ac = 9 - -32 = 41.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (3 ± √41) / 2.
5. Calculate roots: x1 = 4.701562118716424, x2 = -1.7015621187164243.
Final Answe... |
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 +... |
Find the roots of the quadratic equation: 2x^2 + 5x + 9 = 0 | 1. Identify coefficients: a=2, b=5, c=9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 5^2 - 4(2)(9) = 25 - 72 = -47.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-5 ± √47i) / 4.
Final Answer: x = -1.25 ± 1.713913650100261i |
Evaluate sin(45°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(45°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.7071 (exact form depends on root simplification if applicable) |
Multiply the following 2x2 matrices:
A =
[4, 2]
[-1, 0]
B =
[2, 5]
[5, 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) = (4*2) + (2*5) = 18
C12 = (A11 * B12) + (A12 * B22) = (4*5) + (2*1) = 22
C21 = (A21 * B11) + (A22 * B21) = (-1*2) + (0*5) = -2
C22 = (A21 * B12) + (A22 * B22) = (-1*5) + (0*1) = -5
Result Matr... |
Find the derivative of f(x) = (3x + 2)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 3x + 2.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 2. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 2)^2 * 3 = 9(3x +... |
Multiply the following 2x2 matrices:
A =
[-1, 0]
[2, -1]
B =
[-2, -5]
[-1, 3] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-1*-2) + (0*-1) = 2
C12 = (A11 * B12) + (A12 * B22) = (-1*-5) + (0*3) = 5
C21 = (A21 * B11) + (A22 * B21) = (2*-2) + (-1*-1) = -3
C22 = (A21 * B12) + (A22 * B22) = (2*-5) + (-1*3) = -13
Resu... |
In a right-angled triangle, the lengths of the two legs are a = 15 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: 15^2 + 10^2 = c^2
2. Calculate the squares: 225 + 100 = c^2
3. Sum the squares: 325 = c^2
4. Take the square root of both sides: c = √325
Final A... |
Add the fractions 8/7 and 1/3. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 7 and 3, which is 21.
2. Convert 8/7 by multiplying top and bottom by 3: 24/21.
3. Convert 1/3 by multiplying top and bottom by 7: 7/21.
4. Add the numerators: 24 + 7 = 31.
5. The resulting fraction is 31/21.
Final Answer: 31/21 |
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 |
Find the roots of the quadratic equation: 3x^2 + -9x + 1 = 0 | 1. Identify coefficients: a=3, b=-9, c=1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 81 - 12 = 69.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (9 ± √69) / 6.
5. Calculate roots: x1 = 2.8844373104863457, x2 = 0.11556268951365418.
Final Answe... |
Add the fractions 11/8 and 7/15. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 8 and 15, which is 120.
2. Convert 11/8 by multiplying top and bottom by 15: 165/120.
3. Convert 7/15 by multiplying top and bottom by 8: 56/120.
4. Add the numerators: 165 + 56 = 221.
5. The resulting fraction is 221/120.
Final ... |
Add the fractions 12/2 and 10/9. 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 12/2 by multiplying top and bottom by 9: 108/18.
3. Convert 10/9 by multiplying top and bottom by 2: 20/18.
4. Add the numerators: 108 + 20 = 128.
5. The resulting fraction is 128/18.
6. Simplify ... |
Multiply the following 2x2 matrices:
A =
[-4, 0]
[0, -5]
B =
[5, 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) = (-4*5) + (0*-2) = -20
C12 = (A11 * B12) + (A12 * B22) = (-4*1) + (0*-2) = -4
C21 = (A21 * B11) + (A22 * B21) = (0*5) + (-5*-2) = 10
C22 = (A21 * B12) + (A22 * B22) = (0*1) + (-5*-2) = 10
Resu... |
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 |
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 |
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 |
Add the fractions 9/11 and 1/3. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 11 and 3, which is 33.
2. Convert 9/11 by multiplying top and bottom by 3: 27/33.
3. Convert 1/3 by multiplying top and bottom by 11: 11/33.
4. Add the numerators: 27 + 11 = 38.
5. The resulting fraction is 38/33.
Final Answer: 3... |
Compute the quotient and remainder of 63702 divided by 99. Show the long division steps. | To divide 63702 by 99:
1. Determine how many times 99 goes into the leading digits of 63702.
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: 63702 / 99 = 643 with a remainder of 45.
Proof: (643 * 99) ... |
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(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) |
Multiply the following 2x2 matrices:
A =
[-3, -4]
[-1, 4]
B =
[0, 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) = (-3*0) + (-4*-1) = 4
C12 = (A11 * B12) + (A12 * B22) = (-3*3) + (-4*0) = -9
C21 = (A21 * B11) + (A22 * B21) = (-1*0) + (4*-1) = -4
C22 = (A21 * B12) + (A22 * B22) = (-1*3) + (4*0) = -3
Result... |
Add the fractions 2/14 and 13/3. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 14 and 3, which is 42.
2. Convert 2/14 by multiplying top and bottom by 3: 6/42.
3. Convert 13/3 by multiplying top and bottom by 14: 182/42.
4. Add the numerators: 6 + 182 = 188.
5. The resulting fraction is 188/42.
6. Simplify ... |
Calculate the mean, population variance, and standard deviation for the dataset: [11, 14, 10, 15, 7] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (11 + 14 + 10 + 15 + 7) / 5 = 57 / 5 = 11.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-0.4, 2.6, -1.4, 3.6, -4.4]
Squared Differences: [0.16, 6.76, 1.96, 12.96, 19.36]
... |
Compute the quotient and remainder of 23951 divided by 37. Show the long division steps. | To divide 23951 by 37:
1. Determine how many times 37 goes into the leading digits of 23951.
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: 23951 / 37 = 647 with a remainder of 12.
Proof: (647 * 37) ... |
Compute the indefinite integral of f(x) = 8x^4. | To find the indefinite integral ∫ 8x^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: 8 / 5 = 8/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 8x^4 dx = 8/5x^5 + C |
Multiply the following 2x2 matrices:
A =
[4, 2]
[-5, 5]
B =
[-4, -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) = (4*-4) + (2*-1) = -18
C12 = (A11 * B12) + (A12 * B22) = (4*-1) + (2*-2) = -8
C21 = (A21 * B11) + (A22 * B21) = (-5*-4) + (5*-1) = 15
C22 = (A21 * B12) + (A22 * B22) = (-5*-1) + (5*-2) = -5
Re... |
Multiply the following 2x2 matrices:
A =
[-5, 5]
[0, -5]
B =
[2, -3]
[-5, 0] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-5*2) + (5*-5) = -35
C12 = (A11 * B12) + (A12 * B22) = (-5*-3) + (5*0) = 15
C21 = (A21 * B11) + (A22 * B21) = (0*2) + (-5*-5) = 25
C22 = (A21 * B12) + (A22 * B22) = (0*-3) + (-5*0) = 0
Resul... |
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 ... |
Multiply the following 2x2 matrices:
A =
[1, -1]
[3, -3]
B =
[-2, 3]
[5, 0] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (1*-2) + (-1*5) = -7
C12 = (A11 * B12) + (A12 * B22) = (1*3) + (-1*0) = 3
C21 = (A21 * B11) + (A22 * B21) = (3*-2) + (-3*5) = -21
C22 = (A21 * B12) + (A22 * B22) = (3*3) + (-3*0) = 9
Result M... |
Add the fractions 6/4 and 12/2. 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 6/4 by multiplying top and bottom by 1: 6/4.
3. Convert 12/2 by multiplying top and bottom by 2: 24/4.
4. Add the numerators: 6 + 24 = 30.
5. The resulting fraction is 30/4.
6. Simplify by dividing... |
Compute the quotient and remainder of 67957 divided by 58. Show the long division steps. | To divide 67957 by 58:
1. Determine how many times 58 goes into the leading digits of 67957.
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: 67957 / 58 = 1171 with a remainder of 39.
Proof: (1171 * 58... |
Compute the indefinite integral of f(x) = 4x^1. | To find the indefinite integral ∫ 4x^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: 4 / 2 = 2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 4x^1 dx = 2x^2 + C |
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