prompt stringlengths 32 115 | completion stringlengths 167 566 |
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Add the fractions 11/6 and 5/11. 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 11/6 by multiplying top and bottom by 11: 121/66.
3. Convert 5/11 by multiplying top and bottom by 6: 30/66.
4. Add the numerators: 121 + 30 = 151.
5. The resulting fraction is 151/66.
Final Answ... |
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 ... |
Calculate the mean, population variance, and standard deviation for the dataset: [10, 1, 10, 1, 20] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (10 + 1 + 10 + 1 + 20) / 5 = 42 / 5 = 8.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [1.6, -7.4, 1.6, -7.4, 11.6]
Squared Differences: [2.56, 54.76, 2.56, 54.76, 134.56]
... |
Compute the quotient and remainder of 84342 divided by 6. Show the long division steps. | To divide 84342 by 6:
1. Determine how many times 6 goes into the leading digits of 84342.
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: 84342 / 6 = 14057 with a remainder of 0.
Proof: (14057 * 6) +... |
Evaluate the logarithm: log_3(27) | To evaluate log_3(27), we ask the question: '3 raised to what power equals 27?'
Let x be the unknown power: 3^x = 27
Since 3^3 = 27, it follows that x = 3.
Final Answer: 3 |
Find the roots of the quadratic equation: 1x^2 + -6x + 4 = 0 | 1. Identify coefficients: a=1, b=-6, c=4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 36 - 16 = 20.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (6 ± √20) / 2.
5. Calculate roots: x1 = 5.23606797749979, x2 = 0.7639320225002102.
Final Answer: ... |
Evaluate the logarithm: log_3(27) | To evaluate log_3(27), we ask the question: '3 raised to what power equals 27?'
Let x be the unknown power: 3^x = 27
Since 3^3 = 27, it follows that x = 3.
Final Answer: 3 |
Find the roots of the quadratic equation: 9x^2 + 6x + 9 = 0 | 1. Identify coefficients: a=9, b=6, c=9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 6^2 - 4(9)(9) = 36 - 324 = -288.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-6 ± √288i) / 18.
Final Answer: x = -0.3333333333333333 ± 0.9428090415820632i |
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... |
Multiply the following 2x2 matrices:
A =
[-1, 4]
[3, -1]
B =
[1, 3]
[-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) = (-1*1) + (4*-1) = -5
C12 = (A11 * B12) + (A12 * B22) = (-1*3) + (4*-4) = -19
C21 = (A21 * B11) + (A22 * B21) = (3*1) + (-1*-1) = 4
C22 = (A21 * B12) + (A22 * B22) = (3*3) + (-1*-4) = 13
Resul... |
Compute the quotient and remainder of 26590 divided by 16. Show the long division steps. | To divide 26590 by 16:
1. Determine how many times 16 goes into the leading digits of 26590.
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: 26590 / 16 = 1661 with a remainder of 14.
Proof: (1661 * 16... |
Multiply the following 2x2 matrices:
A =
[2, 1]
[3, 4]
B =
[2, -5]
[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) + (1*3) = 7
C12 = (A11 * B12) + (A12 * B22) = (2*-5) + (1*3) = -7
C21 = (A21 * B11) + (A22 * B21) = (3*2) + (4*3) = 18
C22 = (A21 * B12) + (A22 * B22) = (3*-5) + (4*3) = -3
Result Matri... |
Add the fractions 9/4 and 8/4. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 4 and 4, which is 4.
2. Convert 9/4 by multiplying top and bottom by 1: 9/4.
3. Convert 8/4 by multiplying top and bottom by 1: 8/4.
4. Add the numerators: 9 + 8 = 17.
5. The resulting fraction is 17/4.
Final Answer: 17/4 |
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 |
In a right-angled triangle, the lengths of the two legs are a = 14 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: 14^2 + 4^2 = c^2
2. Calculate the squares: 196 + 16 = c^2
3. Sum the squares: 212 = c^2
4. Take the square root of both sides: c = √212
Final Ans... |
Multiply the following 2x2 matrices:
A =
[-1, -5]
[3, 5]
B =
[3, -4]
[3, -4] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-1*3) + (-5*3) = -18
C12 = (A11 * B12) + (A12 * B22) = (-1*-4) + (-5*-4) = 24
C21 = (A21 * B11) + (A22 * B21) = (3*3) + (5*3) = 24
C22 = (A21 * B12) + (A22 * B22) = (3*-4) + (5*-4) = -32
Res... |
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 |
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 ... |
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: [14, 3, 7, 19, 8] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (14 + 3 + 7 + 19 + 8) / 5 = 51 / 5 = 10.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [3.8, -7.2, -3.2, 8.8, -2.2]
Squared Differences: [14.44, 51.84, 10.24, 77.44, 4.84]
... |
Find the roots of the quadratic equation: 10x^2 + -5x + -6 = 0 | 1. Identify coefficients: a=10, b=-5, c=-6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 25 - -240 = 265.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (5 ± √265) / 20.
5. Calculate roots: x1 = 1.0639410298049854, x2 = -0.5639410298049853.
Fina... |
Calculate the mean, population variance, and standard deviation for the dataset: [19, 3, 1, 16, 13] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (19 + 3 + 1 + 16 + 13) / 5 = 52 / 5 = 10.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [8.6, -7.4, -9.4, 5.6, 2.6]
Squared Differences: [73.96, 54.76, 88.36, 31.36, 6.76]
... |
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) |
Find the roots of the quadratic equation: 9x^2 + 3x + 3 = 0 | 1. Identify coefficients: a=9, b=3, c=3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 3^2 - 4(9)(3) = 9 - 108 = -99.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-3 ± √99i) / 18.
Final Answer: x = -0.16666666666666666 ± 0.5527707983925666i |
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 ... |
In a right-angled triangle, the lengths of the two legs are a = 12 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: 12^2 + 6^2 = c^2
2. Calculate the squares: 144 + 36 = c^2
3. Sum the squares: 180 = c^2
4. Take the square root of both sides: c = √180
Final Ans... |
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) = 7x^1. | To find the indefinite integral ∫ 7x^1 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 1 + 1 = 2.
2. Divide the coefficient by the new exponent: 7 / 2 = 7/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 7x^1 dx = 7/2x^2 + C |
Find the 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 +... |
Compute the quotient and remainder of 45179 divided by 15. Show the long division steps. | To divide 45179 by 15:
1. Determine how many times 15 goes into the leading digits of 45179.
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: 45179 / 15 = 3011 with a remainder of 14.
Proof: (3011 * 15... |
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) |
Add the fractions 6/7 and 3/2. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 7 and 2, which is 14.
2. Convert 6/7 by multiplying top and bottom by 2: 12/14.
3. Convert 3/2 by multiplying top and bottom by 7: 21/14.
4. Add the numerators: 12 + 21 = 33.
5. The resulting fraction is 33/14.
Final Answer: 33/1... |
Add the fractions 6/15 and 5/3. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 15 and 3, which is 15.
2. Convert 6/15 by multiplying top and bottom by 1: 6/15.
3. Convert 5/3 by multiplying top and bottom by 5: 25/15.
4. Add the numerators: 6 + 25 = 31.
5. The resulting fraction is 31/15.
Final Answer: 31/1... |
Compute the quotient and remainder of 42972 divided by 28. Show the long division steps. | To divide 42972 by 28:
1. Determine how many times 28 goes into the leading digits of 42972.
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: 42972 / 28 = 1534 with a remainder of 20.
Proof: (1534 * 28... |
Compute the quotient and remainder of 14392 divided by 79. Show the long division steps. | To divide 14392 by 79:
1. Determine how many times 79 goes into the leading digits of 14392.
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: 14392 / 79 = 182 with a remainder of 14.
Proof: (182 * 79) ... |
Multiply the following 2x2 matrices:
A =
[-1, -1]
[3, -5]
B =
[2, 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) = (-1*2) + (-1*4) = -6
C12 = (A11 * B12) + (A12 * B22) = (-1*2) + (-1*-4) = 2
C21 = (A21 * B11) + (A22 * B21) = (3*2) + (-5*4) = -14
C22 = (A21 * B12) + (A22 * B22) = (3*2) + (-5*-4) = 26
Resul... |
Multiply the following 2x2 matrices:
A =
[1, -2]
[-3, -3]
B =
[5, 0]
[-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*5) + (-2*-3) = 11
C12 = (A11 * B12) + (A12 * B22) = (1*0) + (-2*-4) = 8
C21 = (A21 * B11) + (A22 * B21) = (-3*5) + (-3*-3) = -6
C22 = (A21 * B12) + (A22 * B22) = (-3*0) + (-3*-4) = 12
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 |
Find the roots of the quadratic equation: 7x^2 + 3x + 2 = 0 | 1. Identify coefficients: a=7, b=3, c=2.
2. Calculate the discriminant: Δ = b^2 - 4ac = 3^2 - 4(7)(2) = 9 - 56 = -47.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-3 ± √47i) / 14.
Final Answer: x = -0.21428571428571427 ± 0.48968961431436026i |
Find the roots of the quadratic equation: 4x^2 + 8x + -6 = 0 | 1. Identify coefficients: a=4, b=8, c=-6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 64 - -96 = 160.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-8 ± √160) / 8.
5. Calculate roots: x1 = 0.5811388300841898, x2 = -2.58113883008419.
Final Ans... |
Find the roots of the quadratic equation: 5x^2 + 3x + -4 = 0 | 1. Identify coefficients: a=5, b=3, c=-4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 9 - -80 = 89.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-3 ± √89) / 10.
5. Calculate roots: x1 = 0.6433981132056603, x2 = -1.2433981132056604.
Final Ans... |
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 +... |
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 5/3 and 1/6. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 3 and 6, which is 6.
2. Convert 5/3 by multiplying top and bottom by 2: 10/6.
3. Convert 1/6 by multiplying top and bottom by 1: 1/6.
4. Add the numerators: 10 + 1 = 11.
5. The resulting fraction is 11/6.
Final Answer: 11/6 |
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 |
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 =
[-5, 1]
[-4, -3]
B =
[-5, 2]
[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) = (-5*-5) + (1*2) = 27
C12 = (A11 * B12) + (A12 * B22) = (-5*2) + (1*-4) = -14
C21 = (A21 * B11) + (A22 * B21) = (-4*-5) + (-3*2) = 14
C22 = (A21 * B12) + (A22 * B22) = (-4*2) + (-3*-4) = 4
Res... |
In a right-angled triangle, the lengths of the two legs are a = 11 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: 11^2 + 15^2 = c^2
2. Calculate the squares: 121 + 225 = c^2
3. Sum the squares: 346 = c^2
4. Take the square root of both sides: c = √346
Final A... |
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 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 |
Add the fractions 14/2 and 5/2. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 2 and 2, which is 2.
2. Convert 14/2 by multiplying top and bottom by 1: 14/2.
3. Convert 5/2 by multiplying top and bottom by 1: 5/2.
4. Add the numerators: 14 + 5 = 19.
5. The resulting fraction is 19/2.
Final Answer: 19/2 |
Evaluate the logarithm: log_3(27) | To evaluate log_3(27), we ask the question: '3 raised to what power equals 27?'
Let x be the unknown power: 3^x = 27
Since 3^3 = 27, it follows that x = 3.
Final Answer: 3 |
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 ... |
Compute the indefinite integral of f(x) = 5x^4. | To find the indefinite integral ∫ 5x^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: 5 / 5 = 1.
3. Combine and add the constant of integration C.
Final Answer: ∫ 5x^4 dx = 1x^5 + C |
Evaluate tan(0°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(0°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.0000 (exact form depends on root simplification if applicable) |
Compute the indefinite integral of f(x) = 3x^4. | To find the indefinite integral ∫ 3x^4 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 4 + 1 = 5.
2. Divide the coefficient by the new exponent: 3 / 5 = 3/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 3x^4 dx = 3/5x^5 + C |
Find the roots of the quadratic equation: 4x^2 + -10x + -5 = 0 | 1. Identify coefficients: a=4, b=-10, c=-5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 100 - -80 = 180.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (10 ± √180) / 8.
5. Calculate roots: x1 = 2.9270509831248424, x2 = -0.42705098312484235.
Fin... |
Evaluate sin(90°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(90°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 1.0000 (exact form depends on root simplification if applicable) |
Find the roots of the quadratic equation: 2x^2 + -4x + 9 = 0 | 1. Identify coefficients: a=2, b=-4, c=9.
2. Calculate the discriminant: Δ = b^2 - 4ac = -4^2 - 4(2)(9) = 16 - 72 = -56.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (4 ± √56i) / 4.
Final Answer: x = 1.0 ± 1.8708286933869707i |
Calculate the mean, population variance, and standard deviation for the dataset: [9, 1, 15, 10, 1] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (9 + 1 + 15 + 10 + 1) / 5 = 36 / 5 = 7.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [1.8, -6.2, 7.8, 2.8, -6.2]
Squared Differences: [3.24, 38.44, 60.84, 7.84, 38.44]
... |
In a right-angled triangle, the lengths of the two legs are a = 19 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: 19^2 + 13^2 = c^2
2. Calculate the squares: 361 + 169 = c^2
3. Sum the squares: 530 = c^2
4. Take the square root of both sides: c = √530
Final A... |
Find the roots of the quadratic equation: 1x^2 + -6x + 6 = 0 | 1. Identify coefficients: a=1, b=-6, c=6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 36 - 24 = 12.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (6 ± √12) / 2.
5. Calculate roots: x1 = 4.732050807568877, x2 = 1.2679491924311228.
Final Answer:... |
Multiply the following 2x2 matrices:
A =
[5, 5]
[-5, -5]
B =
[-3, 2]
[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) = (5*-3) + (5*4) = 5
C12 = (A11 * B12) + (A12 * B22) = (5*2) + (5*3) = 25
C21 = (A21 * B11) + (A22 * B21) = (-5*-3) + (-5*4) = -5
C22 = (A21 * B12) + (A22 * B22) = (-5*2) + (-5*3) = -25
Result ... |
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 |
Find the roots of the quadratic equation: 1x^2 + -10x + 9 = 0 | 1. Identify coefficients: a=1, b=-10, c=9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 100 - 36 = 64.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (10 ± √64) / 2.
5. Calculate roots: x1 = 9.0, x2 = 1.0.
Final Answer: x = 9.0, 1.0 |
In a right-angled triangle, the lengths of the two legs are a = 20 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: 20^2 + 18^2 = c^2
2. Calculate the squares: 400 + 324 = c^2
3. Sum the squares: 724 = c^2
4. Take the square root of both sides: c = √724
Final A... |
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: [9, 7, 9, 2, 18] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (9 + 7 + 9 + 2 + 18) / 5 = 45 / 5 = 9.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [0.0, -2.0, 0.0, -7.0, 9.0]
Squared Differences: [0.0, 4.0, 0.0, 49.0, 81.0]
Sum of ... |
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 |
Find the roots of the quadratic equation: 9x^2 + 6x + -7 = 0 | 1. Identify coefficients: a=9, b=6, c=-7.
2. Calculate the discriminant: Δ = b^2 - 4ac = 36 - -252 = 288.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-6 ± √288) / 18.
5. Calculate roots: x1 = 0.60947570824873, x2 = -1.2761423749153966.
Final A... |
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 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 =
[1, 3]
[1, 0]
B =
[-1, 1]
[-5, 3] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (1*-1) + (3*-5) = -16
C12 = (A11 * B12) + (A12 * B22) = (1*1) + (3*3) = 10
C21 = (A21 * B11) + (A22 * B21) = (1*-1) + (0*-5) = -1
C22 = (A21 * B12) + (A22 * B22) = (1*1) + (0*3) = 1
Result Ma... |
Add the fractions 9/4 and 15/9. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 4 and 9, which is 36.
2. Convert 9/4 by multiplying top and bottom by 9: 81/36.
3. Convert 15/9 by multiplying top and bottom by 4: 60/36.
4. Add the numerators: 81 + 60 = 141.
5. The resulting fraction is 141/36.
6. Simplify by ... |
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 ... |
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) |
Calculate the mean, population variance, and standard deviation for the dataset: [12, 18, 2, 12, 3] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (12 + 18 + 2 + 12 + 3) / 5 = 47 / 5 = 9.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [2.6, 8.6, -7.4, 2.6, -6.4]
Squared Differences: [6.76, 73.96, 54.76, 6.76, 40.96]
... |
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) |
In a right-angled triangle, the lengths of the two legs are a = 12 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: 12^2 + 13^2 = c^2
2. Calculate the squares: 144 + 169 = c^2
3. Sum the squares: 313 = c^2
4. Take the square root of both sides: c = √313
Final A... |
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 |
Find the roots of the quadratic equation: 1x^2 + 3x + 4 = 0 | 1. Identify coefficients: a=1, b=3, c=4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 3^2 - 4(1)(4) = 9 - 16 = -7.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-3 ± √7i) / 2.
Final Answer: x = -1.5 ± 1.3228756555322954i |
Multiply the following 2x2 matrices:
A =
[-3, 1]
[-2, 0]
B =
[5, -1]
[-1, -2] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-3*5) + (1*-1) = -16
C12 = (A11 * B12) + (A12 * B22) = (-3*-1) + (1*-2) = 1
C21 = (A21 * B11) + (A22 * B21) = (-2*5) + (0*-1) = -10
C22 = (A21 * B12) + (A22 * B22) = (-2*-1) + (0*-2) = 2
Res... |
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 |
Evaluate sin(0°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(0°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.0000 (exact form depends on root simplification if applicable) |
Calculate the mean, population variance, and standard deviation for the dataset: [7, 19, 13, 10, 9] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (7 + 19 + 13 + 10 + 9) / 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, 7.4, 1.4, -1.6, -2.6]
Squared Differences: [21.16, 54.76, 1.96, 2.56, 6.76]
... |
Compute the indefinite integral of f(x) = 5x^4. | To find the indefinite integral ∫ 5x^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: 5 / 5 = 1.
3. Combine and add the constant of integration C.
Final Answer: ∫ 5x^4 dx = 1x^5 + C |
Find the roots of the quadratic equation: 9x^2 + 0x + 2 = 0 | 1. Identify coefficients: a=9, b=0, c=2.
2. Calculate the discriminant: Δ = b^2 - 4ac = 0^2 - 4(9)(2) = 0 - 72 = -72.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (0 ± √72i) / 18.
Final Answer: x = 0.0 ± 0.4714045207910316i |
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) |
In a right-angled triangle, the lengths of the two legs are a = 3 and b = 11. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 3^2 + 11^2 = c^2
2. Calculate the squares: 9 + 121 = c^2
3. Sum the squares: 130 = c^2
4. Take the square root of both sides: c = √130
Final Answ... |
Find the roots of the quadratic equation: 6x^2 + 3x + 4 = 0 | 1. Identify coefficients: a=6, b=3, c=4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 3^2 - 4(6)(4) = 9 - 96 = -87.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-3 ± √87i) / 12.
Final Answer: x = -0.25 ± 0.7772815877574013i |
Find the roots of the quadratic equation: 8x^2 + -10x + -10 = 0 | 1. Identify coefficients: a=8, b=-10, c=-10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 100 - -320 = 420.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (10 ± √420) / 16.
5. Calculate roots: x1 = 1.9058688457449497, x2 = -0.6558688457449497.
F... |
In a right-angled triangle, the lengths of the two legs are a = 9 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: 9^2 + 3^2 = c^2
2. Calculate the squares: 81 + 9 = c^2
3. Sum the squares: 90 = c^2
4. Take the square root of both sides: c = √90
Final Answer: ... |
Add the fractions 12/2 and 6/4. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 2 and 4, which is 4.
2. Convert 12/2 by multiplying top and bottom by 2: 24/4.
3. Convert 6/4 by multiplying top and bottom by 1: 6/4.
4. Add the numerators: 24 + 6 = 30.
5. The resulting fraction is 30/4.
6. Simplify by dividing... |
Calculate the mean, population variance, and standard deviation for the dataset: [2, 9, 7, 18, 9] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (2 + 9 + 7 + 18 + 9) / 5 = 45 / 5 = 9.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-7.0, 0.0, -2.0, 9.0, 0.0]
Squared Differences: [49.0, 0.0, 4.0, 81.0, 0.0]
Sum of ... |
In a right-angled triangle, the lengths of the two legs are a = 13 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: 13^2 + 12^2 = c^2
2. Calculate the squares: 169 + 144 = c^2
3. Sum the squares: 313 = c^2
4. Take the square root of both sides: c = √313
Final A... |
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 |
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 +... |
Multiply the following 2x2 matrices:
A =
[-5, 3]
[3, -3]
B =
[1, -2]
[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) = (-5*1) + (3*1) = -2
C12 = (A11 * B12) + (A12 * B22) = (-5*-2) + (3*3) = 19
C21 = (A21 * B11) + (A22 * B21) = (3*1) + (-3*1) = 0
C22 = (A21 * B12) + (A22 * B22) = (3*-2) + (-3*3) = -15
Result ... |
Multiply the following 2x2 matrices:
A =
[1, -1]
[-1, -3]
B =
[-5, 1]
[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*-5) + (-1*0) = -5
C12 = (A11 * B12) + (A12 * B22) = (1*1) + (-1*2) = -1
C21 = (A21 * B11) + (A22 * B21) = (-1*-5) + (-3*0) = 5
C22 = (A21 * B12) + (A22 * B22) = (-1*1) + (-3*2) = -7
Result... |
Calculate the mean, population variance, and standard deviation for the dataset: [15, 11, 9, 7, 16] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (15 + 11 + 9 + 7 + 16) / 5 = 58 / 5 = 11.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [3.4, -0.6, -2.6, -4.6, 4.4]
Squared Differences: [11.56, 0.36, 6.76, 21.16, 19.36]
... |
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