prompt stringlengths 32 115 | completion stringlengths 167 566 |
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Compute the quotient and remainder of 98415 divided by 32. Show the long division steps. | To divide 98415 by 32:
1. Determine how many times 32 goes into the leading digits of 98415.
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: 98415 / 32 = 3075 with a remainder of 15.
Proof: (3075 * 32... |
Find the roots of the quadratic equation: 3x^2 + 5x + 0 = 0 | 1. Identify coefficients: a=3, b=5, c=0.
2. Calculate the discriminant: Δ = b^2 - 4ac = 25 - 0 = 25.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-5 ± √25) / 6.
5. Calculate roots: x1 = 0.0, x2 = -1.6666666666666667.
Final Answer: x = 0.0, -1.6... |
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 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 roots of the quadratic equation: 9x^2 + -9x + 1 = 0 | 1. Identify coefficients: a=9, b=-9, c=1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 81 - 36 = 45.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (9 ± √45) / 18.
5. Calculate roots: x1 = 0.872677996249965, x2 = 0.12732200375003502.
Final Answe... |
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 +... |
Find the derivative of f(x) = (5x + 4)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 5x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 5x + 4. Its derivative is h'(x) = 5.
4. Multiply g'(u) by h'(x): f'(x) = 3(5x + 4)^2 * 5 = 15(5x ... |
Compute the indefinite integral of f(x) = 3x^5. | To find the indefinite integral ∫ 3x^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: 3 / 6 = 1/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 3x^5 dx = 1/2x^6 + C |
Add the fractions 11/15 and 5/12. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 15 and 12, which is 60.
2. Convert 11/15 by multiplying top and bottom by 4: 44/60.
3. Convert 5/12 by multiplying top and bottom by 5: 25/60.
4. Add the numerators: 44 + 25 = 69.
5. The resulting fraction is 69/60.
6. Simplify b... |
Find the roots of the quadratic equation: 2x^2 + 7x + 10 = 0 | 1. Identify coefficients: a=2, b=7, c=10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 7^2 - 4(2)(10) = 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) / 4.
Final Answer: x = -1.75 ± 1.3919410907075054i |
Add the fractions 5/14 and 10/10. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 14 and 10, which is 70.
2. Convert 5/14 by multiplying top and bottom by 5: 25/70.
3. Convert 10/10 by multiplying top and bottom by 7: 70/70.
4. Add the numerators: 25 + 70 = 95.
5. The resulting fraction is 95/70.
6. Simplify b... |
Find the derivative of f(x) = (5x + 3)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 5x + 3.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 5x + 3. Its derivative is h'(x) = 5.
4. Multiply g'(u) by h'(x): f'(x) = 3(5x + 3)^2 * 5 = 15(5x ... |
Find the 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) = 9x^4. | To find the indefinite integral ∫ 9x^4 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 4 + 1 = 5.
2. Divide the coefficient by the new exponent: 9 / 5 = 9/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 9x^4 dx = 9/5x^5 + C |
Evaluate the logarithm: log_3(243) | To evaluate log_3(243), we ask the question: '3 raised to what power equals 243?'
Let x be the unknown power: 3^x = 243
Since 3^5 = 243, it follows that x = 5.
Final Answer: 5 |
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) |
Multiply the following 2x2 matrices:
A =
[3, -1]
[-5, -2]
B =
[-5, 3]
[-2, 1] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (3*-5) + (-1*-2) = -13
C12 = (A11 * B12) + (A12 * B22) = (3*3) + (-1*1) = 8
C21 = (A21 * B11) + (A22 * B21) = (-5*-5) + (-2*-2) = 29
C22 = (A21 * B12) + (A22 * B22) = (-5*3) + (-2*1) = -17
Re... |
Find the roots of the quadratic equation: 3x^2 + 7x + 5 = 0 | 1. Identify coefficients: a=3, b=7, c=5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 7^2 - 4(3)(5) = 49 - 60 = -11.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-7 ± √11i) / 6.
Final Answer: x = -1.1666666666666667 ± 0.5527707983925666i |
Calculate the mean, population variance, and standard deviation for the dataset: [11, 14, 1, 4, 7] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (11 + 14 + 1 + 4 + 7) / 5 = 37 / 5 = 7.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [3.6, 6.6, -6.4, -3.4, -0.4]
Squared Differences: [12.96, 43.56, 40.96, 11.56, 0.16]
... |
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 |
Calculate the mean, population variance, and standard deviation for the dataset: [8, 13, 12, 18, 9] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (8 + 13 + 12 + 18 + 9) / 5 = 60 / 5 = 12.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-4.0, 1.0, 0.0, 6.0, -3.0]
Squared Differences: [16.0, 1.0, 0.0, 36.0, 9.0]
Sum ... |
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(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: 6x^2 + -4x + -9 = 0 | 1. Identify coefficients: a=6, b=-4, c=-9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 16 - -216 = 232.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (4 ± √232) / 12.
5. Calculate roots: x1 = 1.602628850977318, x2 = -0.9359621843106515.
Final ... |
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 |
Evaluate tan(0°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(0°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.0000 (exact form depends on root simplification if applicable) |
Find the roots of the quadratic equation: 8x^2 + -1x + 3 = 0 | 1. Identify coefficients: a=8, b=-1, c=3.
2. Calculate the discriminant: Δ = b^2 - 4ac = -1^2 - 4(8)(3) = 1 - 96 = -95.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (1 ± √95i) / 16.
Final Answer: x = 0.0625 ± 0.6091746465505602i |
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 ... |
Calculate the mean, population variance, and standard deviation for the dataset: [5, 1, 10, 20, 9] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (5 + 1 + 10 + 20 + 9) / 5 = 45 / 5 = 9.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-4.0, -8.0, 1.0, 11.0, 0.0]
Squared Differences: [16.0, 64.0, 1.0, 121.0, 0.0]
Sum... |
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(30°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(30°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.5774 (exact form depends on root simplification if applicable) |
Compute the quotient and remainder of 65258 divided by 50. Show the long division steps. | To divide 65258 by 50:
1. Determine how many times 50 goes into the leading digits of 65258.
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: 65258 / 50 = 1305 with a remainder of 8.
Proof: (1305 * 50)... |
Multiply the following 2x2 matrices:
A =
[-3, -5]
[5, 1]
B =
[-4, -5]
[-5, -5] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-3*-4) + (-5*-5) = 37
C12 = (A11 * B12) + (A12 * B22) = (-3*-5) + (-5*-5) = 40
C21 = (A21 * B11) + (A22 * B21) = (5*-4) + (1*-5) = -25
C22 = (A21 * B12) + (A22 * B22) = (5*-5) + (1*-5) = -30
... |
Find the roots of the quadratic equation: 7x^2 + 9x + 3 = 0 | 1. Identify coefficients: a=7, b=9, c=3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 9^2 - 4(7)(3) = 81 - 84 = -3.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-9 ± √3i) / 14.
Final Answer: x = -0.6428571428571429 ± 0.12371791482634838i |
In a right-angled triangle, the lengths of the two legs are a = 16 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: 16^2 + 7^2 = c^2
2. Calculate the squares: 256 + 49 = c^2
3. Sum the squares: 305 = c^2
4. Take the square root of both sides: c = √305
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 |
Find the derivative of f(x) = (4x + 3)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 4x + 3.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 3. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 3)^2 * 4 = 12(4x ... |
Evaluate the logarithm: log_2(4) | To evaluate log_2(4), we ask the question: '2 raised to what power equals 4?'
Let x be the unknown power: 2^x = 4
Since 2^2 = 4, it follows that x = 2.
Final Answer: 2 |
Find the derivative of f(x) = (5x + 3)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 5x + 3.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 5x + 3. Its derivative is h'(x) = 5.
4. Multiply g'(u) by h'(x): f'(x) = 3(5x + 3)^2 * 5 = 15(5x ... |
Find the roots of the quadratic equation: 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 ... |
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 +... |
Calculate the mean, population variance, and standard deviation for the dataset: [20, 4, 2, 2, 1] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (20 + 4 + 2 + 2 + 1) / 5 = 29 / 5 = 5.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [14.2, -1.8, -3.8, -3.8, -4.8]
Squared Differences: [201.64, 3.24, 14.44, 14.44, 23.04]... |
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 |
Calculate the mean, population variance, and standard deviation for the dataset: [20, 9, 2, 6, 9] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (20 + 9 + 2 + 6 + 9) / 5 = 46 / 5 = 9.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [10.8, -0.2, -7.2, -3.2, -0.2]
Squared Differences: [116.64, 0.04, 51.84, 10.24, 0.04]
... |
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 |
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 |
In a right-angled triangle, the lengths of the two legs are a = 13 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: 13^2 + 11^2 = c^2
2. Calculate the squares: 169 + 121 = c^2
3. Sum the squares: 290 = c^2
4. Take the square root of both sides: c = √290
Final A... |
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: [15, 10, 10, 2, 14] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (15 + 10 + 10 + 2 + 14) / 5 = 51 / 5 = 10.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [4.8, -0.2, -0.2, -8.2, 3.8]
Squared Differences: [23.04, 0.04, 0.04, 67.24, 14.44]... |
Add the fractions 15/6 and 9/10. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 6 and 10, which is 30.
2. Convert 15/6 by multiplying top and bottom by 5: 75/30.
3. Convert 9/10 by multiplying top and bottom by 3: 27/30.
4. Add the numerators: 75 + 27 = 102.
5. The resulting fraction is 102/30.
6. Simplify b... |
Compute the quotient and remainder of 64998 divided by 65. Show the long division steps. | To divide 64998 by 65:
1. Determine how many times 65 goes into the leading digits of 64998.
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: 64998 / 65 = 999 with a remainder of 63.
Proof: (999 * 65) ... |
Compute the quotient and remainder of 49627 divided by 33. Show the long division steps. | To divide 49627 by 33:
1. Determine how many times 33 goes into the leading digits of 49627.
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: 49627 / 33 = 1503 with a remainder of 28.
Proof: (1503 * 33... |
Compute the quotient and remainder of 41844 divided by 35. Show the long division steps. | To divide 41844 by 35:
1. Determine how many times 35 goes into the leading digits of 41844.
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: 41844 / 35 = 1195 with a remainder of 19.
Proof: (1195 * 35... |
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 +... |
Calculate the mean, population variance, and standard deviation for the dataset: [15, 1, 11, 9, 10] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (15 + 1 + 11 + 9 + 10) / 5 = 46 / 5 = 9.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [5.8, -8.2, 1.8, -0.2, 0.8]
Squared Differences: [33.64, 67.24, 3.24, 0.04, 0.64]
... |
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) |
In a right-angled triangle, the lengths of the two legs are a = 16 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: 16^2 + 5^2 = c^2
2. Calculate the squares: 256 + 25 = c^2
3. Sum the squares: 281 = c^2
4. Take the square root of both sides: c = √281
Final Ans... |
Evaluate tan(0°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(0°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.0000 (exact form depends on root simplification if applicable) |
Find the 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 ... |
Calculate the mean, population variance, and standard deviation for the dataset: [15, 15, 7, 12, 13] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (15 + 15 + 7 + 12 + 13) / 5 = 62 / 5 = 12.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [2.6, 2.6, -5.4, -0.4, 0.6]
Squared Differences: [6.76, 6.76, 29.16, 0.16, 0.36]
... |
Find the roots of the quadratic equation: 1x^2 + 10x + -3 = 0 | 1. Identify coefficients: a=1, b=10, c=-3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 100 - -12 = 112.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-10 ± √112) / 2.
5. Calculate roots: x1 = 0.29150262212918143, x2 = -10.291502622129181.
Fin... |
Find the roots of the quadratic equation: 7x^2 + 8x + -1 = 0 | 1. Identify coefficients: a=7, b=8, c=-1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 64 - -28 = 92.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-8 ± √92) / 14.
5. Calculate roots: x1 = 0.11369021761610273, x2 = -1.2565473604732456.
Final A... |
In a right-angled triangle, the lengths of the two legs are a = 3 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: 3^2 + 12^2 = c^2
2. Calculate the squares: 9 + 144 = c^2
3. Sum the squares: 153 = c^2
4. Take the square root of both sides: c = √153
Final Answ... |
Compute the quotient and remainder of 96229 divided by 99. Show the long division steps. | To divide 96229 by 99:
1. Determine how many times 99 goes into the leading digits of 96229.
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: 96229 / 99 = 972 with a remainder of 1.
Proof: (972 * 99) +... |
Find the roots of the quadratic equation: 6x^2 + -2x + 9 = 0 | 1. Identify coefficients: a=6, b=-2, c=9.
2. Calculate the discriminant: Δ = b^2 - 4ac = -2^2 - 4(6)(9) = 4 - 216 = -212.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (2 ± √212i) / 12.
Final Answer: x = 0.16666666666666666 ± 1.2133516482134197i |
Multiply the following 2x2 matrices:
A =
[-3, 4]
[-4, 2]
B =
[2, -5]
[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) = (-3*2) + (4*1) = -2
C12 = (A11 * B12) + (A12 * B22) = (-3*-5) + (4*1) = 19
C21 = (A21 * B11) + (A22 * B21) = (-4*2) + (2*1) = -6
C22 = (A21 * B12) + (A22 * B22) = (-4*-5) + (2*1) = 22
Result ... |
Compute the quotient and remainder of 35145 divided by 13. Show the long division steps. | To divide 35145 by 13:
1. Determine how many times 13 goes into the leading digits of 35145.
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: 35145 / 13 = 2703 with a remainder of 6.
Proof: (2703 * 13)... |
Calculate the mean, population variance, and standard deviation for the dataset: [19, 11, 12, 15, 13] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (19 + 11 + 12 + 15 + 13) / 5 = 70 / 5 = 14.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [5.0, -3.0, -2.0, 1.0, -1.0]
Squared Differences: [25.0, 9.0, 4.0, 1.0, 1.0]
Su... |
Find the derivative of f(x) = (5x + 4)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 5x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 5x + 4. Its derivative is h'(x) = 5.
4. Multiply g'(u) by h'(x): f'(x) = 3(5x + 4)^2 * 5 = 15(5x ... |
Compute the quotient and remainder of 33857 divided by 91. Show the long division steps. | To divide 33857 by 91:
1. Determine how many times 91 goes into the leading digits of 33857.
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: 33857 / 91 = 372 with a remainder of 5.
Proof: (372 * 91) +... |
Find the derivative of f(x) = (5x + 4)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 5x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 5x + 4. Its derivative is h'(x) = 5.
4. Multiply g'(u) by h'(x): f'(x) = 3(5x + 4)^2 * 5 = 15(5x ... |
Find the roots of the quadratic equation: 5x^2 + -4x + -10 = 0 | 1. Identify coefficients: a=5, b=-4, c=-10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 16 - -200 = 216.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (4 ± √216) / 10.
5. Calculate roots: x1 = 1.8696938456699068, x2 = -1.069693845669907.
Final... |
In a right-angled triangle, the lengths of the two legs are a = 15 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: 15^2 + 5^2 = c^2
2. Calculate the squares: 225 + 25 = c^2
3. Sum the squares: 250 = c^2
4. Take the square root of both sides: c = √250
Final Ans... |
Compute the quotient and remainder of 48370 divided by 64. Show the long division steps. | To divide 48370 by 64:
1. Determine how many times 64 goes into the leading digits of 48370.
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: 48370 / 64 = 755 with a remainder of 50.
Proof: (755 * 64) ... |
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) = 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 |
Add the fractions 13/6 and 7/14. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 6 and 14, which is 42.
2. Convert 13/6 by multiplying top and bottom by 7: 91/42.
3. Convert 7/14 by multiplying top and bottom by 3: 21/42.
4. Add the numerators: 91 + 21 = 112.
5. The resulting fraction is 112/42.
6. Simplify b... |
In a right-angled triangle, the lengths of the two legs are a = 20 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: 20^2 + 10^2 = c^2
2. Calculate the squares: 400 + 100 = c^2
3. Sum the squares: 500 = c^2
4. Take the square root of both sides: c = √500
Final A... |
Find the roots of the quadratic equation: 2x^2 + 4x + 2 = 0 | 1. Identify coefficients: a=2, b=4, c=2.
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) / 4.
5. Calculate roots: x1 = -1.0, x2 = -1.0.
Final Answer: x = -1.0, -1.0 |
Compute the quotient and remainder of 53732 divided by 7. Show the long division steps. | To divide 53732 by 7:
1. Determine how many times 7 goes into the leading digits of 53732.
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: 53732 / 7 = 7676 with a remainder of 0.
Proof: (7676 * 7) + 0... |
Add the fractions 2/2 and 12/8. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 2 and 8, which is 8.
2. Convert 2/2 by multiplying top and bottom by 4: 8/8.
3. Convert 12/8 by multiplying top and bottom by 1: 12/8.
4. Add the numerators: 8 + 12 = 20.
5. The resulting fraction is 20/8.
6. Simplify by dividing... |
Find the roots of the quadratic equation: 3x^2 + -7x + 0 = 0 | 1. Identify coefficients: a=3, b=-7, c=0.
2. Calculate the discriminant: Δ = b^2 - 4ac = 49 - 0 = 49.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (7 ± √49) / 6.
5. Calculate roots: x1 = 2.3333333333333335, x2 = 0.0.
Final Answer: x = 2.33333333... |
Find the roots of the quadratic equation: 4x^2 + -9x + 8 = 0 | 1. Identify coefficients: a=4, b=-9, c=8.
2. Calculate the discriminant: Δ = b^2 - 4ac = -9^2 - 4(4)(8) = 81 - 128 = -47.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (9 ± √47i) / 8.
Final Answer: x = 1.125 ± 0.8569568250501305i |
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) = 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 |
In a right-angled triangle, the lengths of the two legs are a = 14 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: 14^2 + 17^2 = c^2
2. Calculate the squares: 196 + 289 = c^2
3. Sum the squares: 485 = c^2
4. Take the square root of both sides: c = √485
Final A... |
Compute the quotient and remainder of 89146 divided by 84. Show the long division steps. | To divide 89146 by 84:
1. Determine how many times 84 goes into the leading digits of 89146.
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: 89146 / 84 = 1061 with a remainder of 22.
Proof: (1061 * 84... |
Calculate the mean, population variance, and standard deviation for the dataset: [16, 5, 9, 4, 10] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (16 + 5 + 9 + 4 + 10) / 5 = 44 / 5 = 8.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [7.2, -3.8, 0.2, -4.8, 1.2]
Squared Differences: [51.84, 14.44, 0.04, 23.04, 1.44]
... |
Evaluate the logarithm: log_3(243) | To evaluate log_3(243), we ask the question: '3 raised to what power equals 243?'
Let x be the unknown power: 3^x = 243
Since 3^5 = 243, it follows that x = 5.
Final Answer: 5 |
In a right-angled triangle, the lengths of the two legs are a = 17 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: 17^2 + 6^2 = c^2
2. Calculate the squares: 289 + 36 = c^2
3. Sum the squares: 325 = c^2
4. Take the square root of both sides: c = √325
Final Ans... |
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 ... |
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) = 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 |
Calculate the mean, population variance, and standard deviation for the dataset: [19, 17, 4, 14, 5] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (19 + 17 + 4 + 14 + 5) / 5 = 59 / 5 = 11.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [7.2, 5.2, -7.8, 2.2, -6.8]
Squared Differences: [51.84, 27.04, 60.84, 4.84, 46.24]
... |
Multiply the following 2x2 matrices:
A =
[0, -4]
[-1, -4]
B =
[5, 2]
[3, 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) = (0*5) + (-4*3) = -12
C12 = (A11 * B12) + (A12 * B22) = (0*2) + (-4*0) = 0
C21 = (A21 * B11) + (A22 * B21) = (-1*5) + (-4*3) = -17
C22 = (A21 * B12) + (A22 * B22) = (-1*2) + (-4*0) = -2
Result... |
In a right-angled triangle, the lengths of the two legs are a = 7 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: 7^2 + 20^2 = c^2
2. Calculate the squares: 49 + 400 = c^2
3. Sum the squares: 449 = c^2
4. Take the square root of both sides: c = √449
Final Ans... |
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) |
Add the fractions 5/10 and 13/13. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 10 and 13, which is 130.
2. Convert 5/10 by multiplying top and bottom by 13: 65/130.
3. Convert 13/13 by multiplying top and bottom by 10: 130/130.
4. Add the numerators: 65 + 130 = 195.
5. The resulting fraction is 195/130.
6. ... |
Multiply the following 2x2 matrices:
A =
[1, -4]
[3, 0]
B =
[-4, -1]
[0, 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*-4) + (-4*0) = -4
C12 = (A11 * B12) + (A12 * B22) = (1*-1) + (-4*1) = -5
C21 = (A21 * B11) + (A22 * B21) = (3*-4) + (0*0) = -12
C22 = (A21 * B12) + (A22 * B22) = (3*-1) + (0*1) = -3
Result... |
Find the roots of the quadratic equation: 4x^2 + 8x + -7 = 0 | 1. Identify coefficients: a=4, b=8, c=-7.
2. Calculate the discriminant: Δ = b^2 - 4ac = 64 - -112 = 176.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-8 ± √176) / 8.
5. Calculate roots: x1 = 0.6583123951776999, x2 = -2.6583123951777.
Final Ans... |
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