prompt stringlengths 32 115 | completion stringlengths 167 566 |
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Add the fractions 6/11 and 9/9. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 11 and 9, which is 99.
2. Convert 6/11 by multiplying top and bottom by 9: 54/99.
3. Convert 9/9 by multiplying top and bottom by 11: 99/99.
4. Add the numerators: 54 + 99 = 153.
5. The resulting fraction is 153/99.
6. Simplify b... |
Find the roots of the quadratic equation: 7x^2 + 4x + -7 = 0 | 1. Identify coefficients: a=7, b=4, c=-7.
2. Calculate the discriminant: Δ = b^2 - 4ac = 16 - -196 = 212.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-4 ± √212) / 14.
5. Calculate roots: x1 = 0.7543014127543597, x2 = -1.325729984182931.
Final ... |
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) |
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) |
In a right-angled triangle, the lengths of the two legs are a = 8 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: 8^2 + 11^2 = c^2
2. Calculate the squares: 64 + 121 = c^2
3. Sum the squares: 185 = c^2
4. Take the square root of both sides: c = √185
Final Ans... |
In a right-angled triangle, the lengths of the two legs are a = 7 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: 7^2 + 19^2 = c^2
2. Calculate the squares: 49 + 361 = c^2
3. Sum the squares: 410 = c^2
4. Take the square root of both sides: c = √410
Final Ans... |
Compute the indefinite integral of f(x) = 1x^4. | To find the indefinite integral ∫ 1x^4 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 4 + 1 = 5.
2. Divide the coefficient by the new exponent: 1 / 5 = 1/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 1x^4 dx = 1/5x^5 + C |
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) |
Find the roots of the quadratic equation: 4x^2 + 4x + -2 = 0 | 1. Identify coefficients: a=4, b=4, c=-2.
2. Calculate the discriminant: Δ = b^2 - 4ac = 16 - -32 = 48.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-4 ± √48) / 8.
5. Calculate roots: x1 = 0.3660254037844386, x2 = -1.3660254037844386.
Final Ans... |
Calculate the mean, population variance, and standard deviation for the dataset: [10, 10, 19, 6, 10] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (10 + 10 + 19 + 6 + 10) / 5 = 55 / 5 = 11.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-1.0, -1.0, 8.0, -5.0, -1.0]
Squared Differences: [1.0, 1.0, 64.0, 25.0, 1.0]
S... |
Find the derivative of f(x) = (3x + 4)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 3x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 4. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 4)^2 * 3 = 9(3x +... |
In a right-angled triangle, the lengths of the two legs are a = 12 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: 12^2 + 4^2 = c^2
2. Calculate the squares: 144 + 16 = c^2
3. Sum the squares: 160 = c^2
4. Take the square root of both sides: c = √160
Final Ans... |
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 ... |
Multiply the following 2x2 matrices:
A =
[5, 3]
[-5, 0]
B =
[-4, -5]
[-4, 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) = (5*-4) + (3*-4) = -32
C12 = (A11 * B12) + (A12 * B22) = (5*-5) + (3*1) = -22
C21 = (A21 * B11) + (A22 * B21) = (-5*-4) + (0*-4) = 20
C22 = (A21 * B12) + (A22 * B22) = (-5*-5) + (0*1) = 25
Res... |
Find the derivative of f(x) = (2x + 2)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 2x + 2.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 2x + 2. Its derivative is h'(x) = 2.
4. Multiply g'(u) by h'(x): f'(x) = 3(2x + 2)^2 * 2 = 6(2x +... |
Find the roots of the quadratic equation: 1x^2 + 2x + -3 = 0 | 1. Identify coefficients: a=1, b=2, c=-3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 4 - -12 = 16.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-2 ± √16) / 2.
5. Calculate roots: x1 = 1.0, x2 = -3.0.
Final Answer: x = 1.0, -3.0 |
Calculate the mean, population variance, and standard deviation for the dataset: [3, 18, 15, 4, 11] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (3 + 18 + 15 + 4 + 11) / 5 = 51 / 5 = 10.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-7.2, 7.8, 4.8, -6.2, 0.8]
Squared Differences: [51.84, 60.84, 23.04, 38.44, 0.64]
... |
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 cos(30°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(30°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.8660 (exact form depends on root simplification if applicable) |
In a right-angled triangle, the lengths of the two legs are a = 6 and b = 20. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 6^2 + 20^2 = c^2
2. Calculate the squares: 36 + 400 = c^2
3. Sum the squares: 436 = c^2
4. Take the square root of both sides: c = √436
Final Ans... |
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 12266 divided by 32. Show the long division steps. | To divide 12266 by 32:
1. Determine how many times 32 goes into the leading digits of 12266.
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: 12266 / 32 = 383 with a remainder of 10.
Proof: (383 * 32) ... |
Calculate the mean, population variance, and standard deviation for the dataset: [10, 13, 16, 10, 14] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (10 + 13 + 16 + 10 + 14) / 5 = 63 / 5 = 12.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-2.6, 0.4, 3.4, -2.6, 1.4]
Squared Differences: [6.76, 0.16, 11.56, 6.76, 1.96]
... |
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) |
Multiply the following 2x2 matrices:
A =
[-1, 1]
[-5, -4]
B =
[-1, 2]
[-2, 1] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-1*-1) + (1*-2) = -1
C12 = (A11 * B12) + (A12 * B22) = (-1*2) + (1*1) = -1
C21 = (A21 * B11) + (A22 * B21) = (-5*-1) + (-4*-2) = 13
C22 = (A21 * B12) + (A22 * B22) = (-5*2) + (-4*1) = -14
Re... |
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 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) = 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 |
Compute the quotient and remainder of 36504 divided by 60. Show the long division steps. | To divide 36504 by 60:
1. Determine how many times 60 goes into the leading digits of 36504.
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: 36504 / 60 = 608 with a remainder of 24.
Proof: (608 * 60) ... |
Multiply the following 2x2 matrices:
A =
[2, -1]
[3, 5]
B =
[1, 3]
[-4, 5] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (2*1) + (-1*-4) = 6
C12 = (A11 * B12) + (A12 * B22) = (2*3) + (-1*5) = 1
C21 = (A21 * B11) + (A22 * B21) = (3*1) + (5*-4) = -17
C22 = (A21 * B12) + (A22 * B22) = (3*3) + (5*5) = 34
Result Mat... |
In a right-angled triangle, the lengths of the two legs are a = 8 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: 8^2 + 11^2 = c^2
2. Calculate the squares: 64 + 121 = c^2
3. Sum the squares: 185 = c^2
4. Take the square root of both sides: c = √185
Final Ans... |
Find the derivative of f(x) = (2x + 3)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 2x + 3.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 2x + 3. Its derivative is h'(x) = 2.
4. Multiply g'(u) by h'(x): f'(x) = 3(2x + 3)^2 * 2 = 6(2x +... |
Find the roots of the quadratic equation: 4x^2 + -5x + 3 = 0 | 1. Identify coefficients: a=4, b=-5, c=3.
2. Calculate the discriminant: Δ = b^2 - 4ac = -5^2 - 4(4)(3) = 25 - 48 = -23.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (5 ± √23i) / 8.
Final Answer: x = 0.625 ± 0.5994789404140899i |
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) |
In a right-angled triangle, the lengths of the two legs are a = 17 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: 17^2 + 5^2 = c^2
2. Calculate the squares: 289 + 25 = c^2
3. Sum the squares: 314 = c^2
4. Take the square root of both sides: c = √314
Final Ans... |
Find the derivative of f(x) = (5x + 2)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 5x + 2.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 5x + 2. Its derivative is h'(x) = 5.
4. Multiply g'(u) by h'(x): f'(x) = 3(5x + 2)^2 * 5 = 15(5x ... |
Find the roots of the quadratic equation: 3x^2 + 3x + 4 = 0 | 1. Identify coefficients: a=3, b=3, c=4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 3^2 - 4(3)(4) = 9 - 48 = -39.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-3 ± √39i) / 6.
Final Answer: x = -0.5 ± 1.0408329997330663i |
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 9/13 and 12/12. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 13 and 12, which is 156.
2. Convert 9/13 by multiplying top and bottom by 12: 108/156.
3. Convert 12/12 by multiplying top and bottom by 13: 156/156.
4. Add the numerators: 108 + 156 = 264.
5. The resulting fraction is 264/156.
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 |
Multiply the following 2x2 matrices:
A =
[-1, -4]
[5, 3]
B =
[-5, 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) = (-1*-5) + (-4*-1) = 9
C12 = (A11 * B12) + (A12 * B22) = (-1*2) + (-4*-1) = 2
C21 = (A21 * B11) + (A22 * B21) = (5*-5) + (3*-1) = -28
C22 = (A21 * B12) + (A22 * B22) = (5*2) + (3*-1) = 7
Resul... |
Compute the quotient and remainder of 82842 divided by 40. Show the long division steps. | To divide 82842 by 40:
1. Determine how many times 40 goes into the leading digits of 82842.
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: 82842 / 40 = 2071 with a remainder of 2.
Proof: (2071 * 40)... |
Compute the indefinite integral of f(x) = 10x^5. | To find the indefinite integral ∫ 10x^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: 10 / 6 = 5/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 10x^5 dx = 5/3x^6 + C |
Multiply the following 2x2 matrices:
A =
[4, -4]
[-4, 0]
B =
[-2, 5]
[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) = (4*-2) + (-4*4) = -24
C12 = (A11 * B12) + (A12 * B22) = (4*5) + (-4*-2) = 28
C21 = (A21 * B11) + (A22 * B21) = (-4*-2) + (0*4) = 8
C22 = (A21 * B12) + (A22 * B22) = (-4*5) + (0*-2) = -20
Resu... |
In a right-angled triangle, the lengths of the two legs are a = 19 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: 19^2 + 5^2 = c^2
2. Calculate the squares: 361 + 25 = c^2
3. Sum the squares: 386 = c^2
4. Take the square root of both sides: c = √386
Final Ans... |
Compute the quotient and remainder of 3764 divided by 42. Show the long division steps. | To divide 3764 by 42:
1. Determine how many times 42 goes into the leading digits of 3764.
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: 3764 / 42 = 89 with a remainder of 26.
Proof: (89 * 42) + 26 ... |
Compute the quotient and remainder of 40722 divided by 35. Show the long division steps. | To divide 40722 by 35:
1. Determine how many times 35 goes into the leading digits of 40722.
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: 40722 / 35 = 1163 with a remainder of 17.
Proof: (1163 * 35... |
Calculate the mean, population variance, and standard deviation for the dataset: [19, 13, 8, 5, 7] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (19 + 13 + 8 + 5 + 7) / 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, 2.6, -2.4, -5.4, -3.4]
Squared Differences: [73.96, 6.76, 5.76, 29.16, 11.56]
... |
In a right-angled triangle, the lengths of the two legs are a = 12 and b = 17. 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 + 17^2 = c^2
2. Calculate the squares: 144 + 289 = c^2
3. Sum the squares: 433 = c^2
4. Take the square root of both sides: c = √433
Final A... |
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) |
Find the roots of the quadratic equation: 5x^2 + 8x + -1 = 0 | 1. Identify coefficients: a=5, b=8, c=-1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 64 - -20 = 84.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-8 ± √84) / 10.
5. Calculate roots: x1 = 0.11651513899116797, x2 = -1.7165151389911677.
Final A... |
In a right-angled triangle, the lengths of the two legs are a = 6 and b = 17. 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: 6^2 + 17^2 = c^2
2. Calculate the squares: 36 + 289 = c^2
3. Sum the squares: 325 = c^2
4. Take the square root of both sides: c = √325
Final Ans... |
Calculate the mean, population variance, and standard deviation for the dataset: [5, 12, 8, 3, 7] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (5 + 12 + 8 + 3 + 7) / 5 = 35 / 5 = 7.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-2.0, 5.0, 1.0, -4.0, 0.0]
Squared Differences: [4.0, 25.0, 1.0, 16.0, 0.0]
Sum of ... |
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 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) |
Calculate the mean, population variance, and standard deviation for the dataset: [10, 20, 18, 18, 8] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (10 + 20 + 18 + 18 + 8) / 5 = 74 / 5 = 14.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-4.8, 5.2, 3.2, 3.2, -6.8]
Squared Differences: [23.04, 27.04, 10.24, 10.24, 46.24... |
Compute the indefinite integral of f(x) = 7x^4. | To find the indefinite integral ∫ 7x^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: 7 / 5 = 7/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 7x^4 dx = 7/5x^5 + C |
Add the fractions 8/13 and 4/13. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 13 and 13, which is 13.
2. Convert 8/13 by multiplying top and bottom by 1: 8/13.
3. Convert 4/13 by multiplying top and bottom by 1: 4/13.
4. Add the numerators: 8 + 4 = 12.
5. The resulting fraction is 12/13.
Final Answer: 12/1... |
Evaluate cos(45°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(45°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.7071 (exact form depends on root simplification if applicable) |
Add the fractions 2/15 and 5/13. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 15 and 13, which is 195.
2. Convert 2/15 by multiplying top and bottom by 13: 26/195.
3. Convert 5/13 by multiplying top and bottom by 15: 75/195.
4. Add the numerators: 26 + 75 = 101.
5. The resulting fraction is 101/195.
Final ... |
Find the derivative of f(x) = (5x + 2)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 5x + 2.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 5x + 2. Its derivative is h'(x) = 5.
4. Multiply g'(u) by h'(x): f'(x) = 3(5x + 2)^2 * 5 = 15(5x ... |
Evaluate the logarithm: log_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 |
Calculate the mean, population variance, and standard deviation for the dataset: [13, 15, 17, 9, 10] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (13 + 15 + 17 + 9 + 10) / 5 = 64 / 5 = 12.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [0.2, 2.2, 4.2, -3.8, -2.8]
Squared Differences: [0.04, 4.84, 17.64, 14.44, 7.84]
... |
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... |
Multiply the following 2x2 matrices:
A =
[-3, -2]
[0, -2]
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) = (-3*2) + (-2*-1) = -4
C12 = (A11 * B12) + (A12 * B22) = (-3*5) + (-2*3) = -21
C21 = (A21 * B11) + (A22 * B21) = (0*2) + (-2*-1) = 2
C22 = (A21 * B12) + (A22 * B22) = (0*5) + (-2*3) = -6
Resul... |
Add the fractions 5/6 and 14/11. 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 5/6 by multiplying top and bottom by 11: 55/66.
3. Convert 14/11 by multiplying top and bottom by 6: 84/66.
4. Add the numerators: 55 + 84 = 139.
5. The resulting fraction is 139/66.
Final Answer... |
Compute the indefinite integral of f(x) = 1x^4. | To find the indefinite integral ∫ 1x^4 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 4 + 1 = 5.
2. Divide the coefficient by the new exponent: 1 / 5 = 1/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 1x^4 dx = 1/5x^5 + C |
Calculate the mean, population variance, and standard deviation for the dataset: [20, 8, 3, 19, 3] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (20 + 8 + 3 + 19 + 3) / 5 = 53 / 5 = 10.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [9.4, -2.6, -7.6, 8.4, -7.6]
Squared Differences: [88.36, 6.76, 57.76, 70.56, 57.76]
... |
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 =
[0, 4]
[-2, -3]
B =
[1, -2]
[5, -2] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (0*1) + (4*5) = 20
C12 = (A11 * B12) + (A12 * B22) = (0*-2) + (4*-2) = -8
C21 = (A21 * B11) + (A22 * B21) = (-2*1) + (-3*5) = -17
C22 = (A21 * B12) + (A22 * B22) = (-2*-2) + (-3*-2) = 10
Resu... |
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 |
Compute the indefinite integral of f(x) = 1x^3. | To find the indefinite integral ∫ 1x^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: 1 / 4 = 1/4.
3. Combine and add the constant of integration C.
Final Answer: ∫ 1x^3 dx = 1/4x^4 + C |
In a right-angled triangle, the lengths of the two legs are a = 3 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: 3^2 + 9^2 = c^2
2. Calculate the squares: 9 + 81 = c^2
3. Sum the squares: 90 = c^2
4. Take the square root of both sides: c = √90
Final Answer: ... |
Compute the indefinite integral of f(x) = 8x^2. | To find the indefinite integral ∫ 8x^2 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 2 + 1 = 3.
2. Divide the coefficient by the new exponent: 8 / 3 = 8/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 8x^2 dx = 8/3x^3 + C |
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 |
Multiply the following 2x2 matrices:
A =
[1, -5]
[3, -3]
B =
[2, -2]
[3, -3] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (1*2) + (-5*3) = -13
C12 = (A11 * B12) + (A12 * B22) = (1*-2) + (-5*-3) = 13
C21 = (A21 * B11) + (A22 * B21) = (3*2) + (-3*3) = -3
C22 = (A21 * B12) + (A22 * B22) = (3*-2) + (-3*-3) = 3
Resul... |
Calculate the mean, population variance, and standard deviation for the dataset: [1, 3, 19, 9, 11] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (1 + 3 + 19 + 9 + 11) / 5 = 43 / 5 = 8.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-7.6, -5.6, 10.4, 0.4, 2.4]
Squared Differences: [57.76, 31.36, 108.16, 0.16, 5.76]
... |
Add the fractions 15/12 and 11/14. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 12 and 14, which is 84.
2. Convert 15/12 by multiplying top and bottom by 7: 105/84.
3. Convert 11/14 by multiplying top and bottom by 6: 66/84.
4. Add the numerators: 105 + 66 = 171.
5. The resulting fraction is 171/84.
6. Simpl... |
Compute the quotient and remainder of 13186 divided by 51. Show the long division steps. | To divide 13186 by 51:
1. Determine how many times 51 goes into the leading digits of 13186.
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: 13186 / 51 = 258 with a remainder of 28.
Proof: (258 * 51) ... |
Compute the indefinite integral of f(x) = 6x^3. | To find the indefinite integral ∫ 6x^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: 6 / 4 = 3/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 6x^3 dx = 3/2x^4 + C |
Add the fractions 5/13 and 13/12. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 13 and 12, which is 156.
2. Convert 5/13 by multiplying top and bottom by 12: 60/156.
3. Convert 13/12 by multiplying top and bottom by 13: 169/156.
4. Add the numerators: 60 + 169 = 229.
5. The resulting fraction is 229/156.
Fin... |
Compute the indefinite integral of f(x) = 9x^1. | To find the indefinite integral ∫ 9x^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: 9 / 2 = 9/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 9x^1 dx = 9/2x^2 + C |
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 |
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 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) |
In a right-angled triangle, the lengths of the two legs are a = 13 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: 13^2 + 8^2 = c^2
2. Calculate the squares: 169 + 64 = c^2
3. Sum the squares: 233 = c^2
4. Take the square root of both sides: c = √233
Final Ans... |
Compute the quotient and remainder of 40608 divided by 78. Show the long division steps. | To divide 40608 by 78:
1. Determine how many times 78 goes into the leading digits of 40608.
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: 40608 / 78 = 520 with a remainder of 48.
Proof: (520 * 78) ... |
In a right-angled triangle, the lengths of the two legs are a = 11 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: 11^2 + 8^2 = c^2
2. Calculate the squares: 121 + 64 = c^2
3. Sum the squares: 185 = c^2
4. Take the square root of both sides: c = √185
Final Ans... |
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) |
In a right-angled triangle, the lengths of the two legs are a = 20 and b = 16. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 20^2 + 16^2 = c^2
2. Calculate the squares: 400 + 256 = c^2
3. Sum the squares: 656 = c^2
4. Take the square root of both sides: c = √656
Final A... |
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 |
Compute the quotient and remainder of 31550 divided by 18. Show the long division steps. | To divide 31550 by 18:
1. Determine how many times 18 goes into the leading digits of 31550.
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: 31550 / 18 = 1752 with a remainder of 14.
Proof: (1752 * 18... |
Find the roots of the quadratic equation: 9x^2 + -8x + -8 = 0 | 1. Identify coefficients: a=9, b=-8, c=-8.
2. Calculate the discriminant: Δ = b^2 - 4ac = 64 - -288 = 352.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (8 ± √352) / 18.
5. Calculate roots: x1 = 1.48675905773854, x2 = -0.597870168849651.
Final An... |
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 |
Find the roots of the quadratic equation: 8x^2 + 2x + -4 = 0 | 1. Identify coefficients: a=8, b=2, c=-4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 4 - -128 = 132.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-2 ± √132) / 16.
5. Calculate roots: x1 = 0.5930703308172536, x2 = -0.8430703308172536.
Final ... |
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 ... |
Add the fractions 14/3 and 5/5. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 3 and 5, which is 15.
2. Convert 14/3 by multiplying top and bottom by 5: 70/15.
3. Convert 5/5 by multiplying top and bottom by 3: 15/15.
4. Add the numerators: 70 + 15 = 85.
5. The resulting fraction is 85/15.
6. Simplify by di... |
Evaluate sin(30°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(30°) 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) = 10x^3. | To find the indefinite integral ∫ 10x^3 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 3 + 1 = 4.
2. Divide the coefficient by the new exponent: 10 / 4 = 5/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 10x^3 dx = 5/2x^4 + C |
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 |
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