prompt stringlengths 32 115 | completion stringlengths 167 566 |
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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 = 4 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: 4^2 + 7^2 = c^2
2. Calculate the squares: 16 + 49 = c^2
3. Sum the squares: 65 = c^2
4. Take the square root of both sides: c = √65
Final Answer:... |
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) = 7x^5. | To find the indefinite integral ∫ 7x^5 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 5 + 1 = 6.
2. Divide the coefficient by the new exponent: 7 / 6 = 7/6.
3. Combine and add the constant of integration C.
Final Answer: ∫ 7x^5 dx = 7/6x^6 + C |
Multiply the following 2x2 matrices:
A =
[4, 1]
[-1, 1]
B =
[1, -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) = (4*1) + (1*2) = 6
C12 = (A11 * B12) + (A12 * B22) = (4*-3) + (1*-1) = -13
C21 = (A21 * B11) + (A22 * B21) = (-1*1) + (1*2) = 1
C22 = (A21 * B12) + (A22 * B22) = (-1*-3) + (1*-1) = 2
Result Ma... |
Multiply the following 2x2 matrices:
A =
[-4, 1]
[-4, -4]
B =
[1, 2]
[-5, -5] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-4*1) + (1*-5) = -9
C12 = (A11 * B12) + (A12 * B22) = (-4*2) + (1*-5) = -13
C21 = (A21 * B11) + (A22 * B21) = (-4*1) + (-4*-5) = 16
C22 = (A21 * B12) + (A22 * B22) = (-4*2) + (-4*-5) = 12
Re... |
Add the fractions 9/8 and 10/2. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 8 and 2, which is 8.
2. Convert 9/8 by multiplying top and bottom by 1: 9/8.
3. Convert 10/2 by multiplying top and bottom by 4: 40/8.
4. Add the numerators: 9 + 40 = 49.
5. The resulting fraction is 49/8.
Final Answer: 49/8 |
Evaluate the logarithm: log_5(125) | To evaluate log_5(125), we ask the question: '5 raised to what power equals 125?'
Let x be the unknown power: 5^x = 125
Since 5^3 = 125, it follows that x = 3.
Final Answer: 3 |
Multiply the following 2x2 matrices:
A =
[-2, 0]
[-3, 1]
B =
[-5, 0]
[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) = (-2*-5) + (0*5) = 10
C12 = (A11 * B12) + (A12 * B22) = (-2*0) + (0*5) = 0
C21 = (A21 * B11) + (A22 * B21) = (-3*-5) + (1*5) = 20
C22 = (A21 * B12) + (A22 * B22) = (-3*0) + (1*5) = 5
Result Ma... |
Compute the quotient and remainder of 36071 divided by 31. Show the long division steps. | To divide 36071 by 31:
1. Determine how many times 31 goes into the leading digits of 36071.
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: 36071 / 31 = 1163 with a remainder of 18.
Proof: (1163 * 31... |
Add the fractions 4/6 and 8/14. 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 4/6 by multiplying top and bottom by 7: 28/42.
3. Convert 8/14 by multiplying top and bottom by 3: 24/42.
4. Add the numerators: 28 + 24 = 52.
5. The resulting fraction is 52/42.
6. Simplify by d... |
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) |
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) |
Add the fractions 7/14 and 11/14. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 14 and 14, which is 14.
2. Convert 7/14 by multiplying top and bottom by 1: 7/14.
3. Convert 11/14 by multiplying top and bottom by 1: 11/14.
4. Add the numerators: 7 + 11 = 18.
5. The resulting fraction is 18/14.
6. Simplify by ... |
Add the fractions 11/9 and 8/12. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 9 and 12, which is 36.
2. Convert 11/9 by multiplying top and bottom by 4: 44/36.
3. Convert 8/12 by multiplying top and bottom by 3: 24/36.
4. Add the numerators: 44 + 24 = 68.
5. The resulting fraction is 68/36.
6. Simplify by ... |
Calculate the mean, population variance, and standard deviation for the dataset: [18, 20, 10, 6, 13] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (18 + 20 + 10 + 6 + 13) / 5 = 67 / 5 = 13.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [4.6, 6.6, -3.4, -7.4, -0.4]
Squared Differences: [21.16, 43.56, 11.56, 54.76, 0.16... |
Find the roots of the quadratic equation: 3x^2 + -10x + -3 = 0 | 1. Identify coefficients: a=3, b=-10, c=-3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 100 - -36 = 136.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (10 ± √136) / 6.
5. Calculate roots: x1 = 3.610317298281767, x2 = -0.2769839649484336.
Final... |
Compute the quotient and remainder of 82785 divided by 80. Show the long division steps. | To divide 82785 by 80:
1. Determine how many times 80 goes into the leading digits of 82785.
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: 82785 / 80 = 1034 with a remainder of 65.
Proof: (1034 * 80... |
Evaluate cos(90°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(90°) 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: [13, 9, 4, 2, 13] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (13 + 9 + 4 + 2 + 13) / 5 = 41 / 5 = 8.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [4.8, 0.8, -4.2, -6.2, 4.8]
Squared Differences: [23.04, 0.64, 17.64, 38.44, 23.04]
... |
Find the roots of the quadratic equation: 3x^2 + -10x + -4 = 0 | 1. Identify coefficients: a=3, b=-10, c=-4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 100 - -48 = 148.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (10 ± √148) / 6.
5. Calculate roots: x1 = 3.694254176766073, x2 = -0.3609208434327398.
Final... |
Evaluate cos(90°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(90°) 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 quotient and remainder of 73919 divided by 16. Show the long division steps. | To divide 73919 by 16:
1. Determine how many times 16 goes into the leading digits of 73919.
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: 73919 / 16 = 4619 with a remainder of 15.
Proof: (4619 * 16... |
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: 6x^2 + 8x + 1 = 0 | 1. Identify coefficients: a=6, b=8, c=1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 64 - 24 = 40.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-8 ± √40) / 12.
5. Calculate roots: x1 = -0.13962038997193674, x2 = -1.1937129433613967.
Final An... |
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 ... |
Find the roots of the quadratic equation: 8x^2 + -4x + -3 = 0 | 1. Identify coefficients: a=8, b=-4, c=-3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 16 - -96 = 112.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (4 ± √112) / 16.
5. Calculate roots: x1 = 0.9114378277661477, x2 = -0.4114378277661477.
Final ... |
Compute the quotient and remainder of 52184 divided by 75. Show the long division steps. | To divide 52184 by 75:
1. Determine how many times 75 goes into the leading digits of 52184.
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: 52184 / 75 = 695 with a remainder of 59.
Proof: (695 * 75) ... |
Add the fractions 12/4 and 6/13. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 4 and 13, which is 52.
2. Convert 12/4 by multiplying top and bottom by 13: 156/52.
3. Convert 6/13 by multiplying top and bottom by 4: 24/52.
4. Add the numerators: 156 + 24 = 180.
5. The resulting fraction is 180/52.
6. Simplif... |
Compute the indefinite integral of f(x) = 3x^1. | To find the indefinite integral ∫ 3x^1 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 1 + 1 = 2.
2. Divide the coefficient by the new exponent: 3 / 2 = 3/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 3x^1 dx = 3/2x^2 + C |
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) |
Add the fractions 6/14 and 8/14. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 14 and 14, which is 14.
2. Convert 6/14 by multiplying top and bottom by 1: 6/14.
3. Convert 8/14 by multiplying top and bottom by 1: 8/14.
4. Add the numerators: 6 + 8 = 14.
5. The resulting fraction is 14/14.
6. Simplify by div... |
In a right-angled triangle, the lengths of the two legs are a = 11 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: 11^2 + 18^2 = c^2
2. Calculate the squares: 121 + 324 = c^2
3. Sum the squares: 445 = c^2
4. Take the square root of both sides: c = √445
Final A... |
Calculate the mean, population variance, and standard deviation for the dataset: [14, 15, 1, 20, 3] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (14 + 15 + 1 + 20 + 3) / 5 = 53 / 5 = 10.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [3.4, 4.4, -9.6, 9.4, -7.6]
Squared Differences: [11.56, 19.36, 92.16, 88.36, 57.76]... |
Compute the indefinite integral of f(x) = 6x^1. | To find the indefinite integral ∫ 6x^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: 6 / 2 = 3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 6x^1 dx = 3x^2 + C |
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 |
Add the fractions 15/5 and 5/4. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 5 and 4, which is 20.
2. Convert 15/5 by multiplying top and bottom by 4: 60/20.
3. Convert 5/4 by multiplying top and bottom by 5: 25/20.
4. Add the numerators: 60 + 25 = 85.
5. The resulting fraction is 85/20.
6. Simplify by di... |
Compute the indefinite integral of f(x) = 4x^5. | To find the indefinite integral ∫ 4x^5 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 5 + 1 = 6.
2. Divide the coefficient by the new exponent: 4 / 6 = 2/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 4x^5 dx = 2/3x^6 + C |
Calculate the mean, population variance, and standard deviation for the dataset: [4, 12, 8, 18, 8] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (4 + 12 + 8 + 18 + 8) / 5 = 50 / 5 = 10.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-6.0, 2.0, -2.0, 8.0, -2.0]
Squared Differences: [36.0, 4.0, 4.0, 64.0, 4.0]
Sum ... |
Calculate the mean, population variance, and standard deviation for the dataset: [3, 7, 19, 10, 6] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (3 + 7 + 19 + 10 + 6) / 5 = 45 / 5 = 9.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-6.0, -2.0, 10.0, 1.0, -3.0]
Squared Differences: [36.0, 4.0, 100.0, 1.0, 9.0]
Sum... |
Add the fractions 1/13 and 5/8. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 13 and 8, which is 104.
2. Convert 1/13 by multiplying top and bottom by 8: 8/104.
3. Convert 5/8 by multiplying top and bottom by 13: 65/104.
4. Add the numerators: 8 + 65 = 73.
5. The resulting fraction is 73/104.
Final Answer:... |
Compute the quotient and remainder of 99684 divided by 70. Show the long division steps. | To divide 99684 by 70:
1. Determine how many times 70 goes into the leading digits of 99684.
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: 99684 / 70 = 1424 with a remainder of 4.
Proof: (1424 * 70)... |
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 ... |
Add the fractions 5/12 and 8/3. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 12 and 3, which is 12.
2. Convert 5/12 by multiplying top and bottom by 1: 5/12.
3. Convert 8/3 by multiplying top and bottom by 4: 32/12.
4. Add the numerators: 5 + 32 = 37.
5. The resulting fraction is 37/12.
Final Answer: 37/1... |
Evaluate the logarithm: log_4(64) | To evaluate log_4(64), we ask the question: '4 raised to what power equals 64?'
Let x be the unknown power: 4^x = 64
Since 4^3 = 64, it follows that x = 3.
Final Answer: 3 |
Multiply the following 2x2 matrices:
A =
[3, 3]
[-5, 3]
B =
[3, 5]
[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) = (3*3) + (3*5) = 24
C12 = (A11 * B12) + (A12 * B22) = (3*5) + (3*-2) = 9
C21 = (A21 * B11) + (A22 * B21) = (-5*3) + (3*5) = 0
C22 = (A21 * B12) + (A22 * B22) = (-5*5) + (3*-2) = -31
Result Mat... |
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 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 |
Add the fractions 10/13 and 13/7. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 13 and 7, which is 91.
2. Convert 10/13 by multiplying top and bottom by 7: 70/91.
3. Convert 13/7 by multiplying top and bottom by 13: 169/91.
4. Add the numerators: 70 + 169 = 239.
5. The resulting fraction is 239/91.
Final Ans... |
Add the fractions 11/8 and 12/2. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 8 and 2, which is 8.
2. Convert 11/8 by multiplying top and bottom by 1: 11/8.
3. Convert 12/2 by multiplying top and bottom by 4: 48/8.
4. Add the numerators: 11 + 48 = 59.
5. The resulting fraction is 59/8.
Final Answer: 59/8 |
Find the roots of the quadratic equation: 5x^2 + 0x + -10 = 0 | 1. Identify coefficients: a=5, b=0, c=-10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 0 - -200 = 200.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (0 ± √200) / 10.
5. Calculate roots: x1 = 1.4142135623730951, x2 = -1.4142135623730951.
Final ... |
Calculate the mean, population variance, and standard deviation for the dataset: [17, 17, 6, 20, 18] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (17 + 17 + 6 + 20 + 18) / 5 = 78 / 5 = 15.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [1.4, 1.4, -9.6, 4.4, 2.4]
Squared Differences: [1.96, 1.96, 92.16, 19.36, 5.76]
... |
Add the fractions 8/8 and 12/9. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 8 and 9, which is 72.
2. Convert 8/8 by multiplying top and bottom by 9: 72/72.
3. Convert 12/9 by multiplying top and bottom by 8: 96/72.
4. Add the numerators: 72 + 96 = 168.
5. The resulting fraction is 168/72.
6. Simplify by ... |
Find the roots of the quadratic equation: 8x^2 + 8x + 7 = 0 | 1. Identify coefficients: a=8, b=8, c=7.
2. Calculate the discriminant: Δ = b^2 - 4ac = 8^2 - 4(8)(7) = 64 - 224 = -160.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-8 ± √160i) / 16.
Final Answer: x = -0.5 ± 0.7905694150420949i |
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 = 19 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: 19^2 + 15^2 = c^2
2. Calculate the squares: 361 + 225 = c^2
3. Sum the squares: 586 = c^2
4. Take the square root of both sides: c = √586
Final A... |
Find the derivative of f(x) = (4x + 3)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 4x + 3.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 3. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 3)^2 * 4 = 12(4x ... |
Compute the indefinite integral of f(x) = 10x^1. | To find the indefinite integral ∫ 10x^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: 10 / 2 = 5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 10x^1 dx = 5x^2 + C |
Find the roots of the quadratic equation: 8x^2 + -9x + 1 = 0 | 1. Identify coefficients: a=8, b=-9, c=1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 81 - 32 = 49.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (9 ± √49) / 16.
5. Calculate roots: x1 = 1.0, x2 = 0.125.
Final Answer: x = 1.0, 0.125 |
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 roots of the quadratic equation: 9x^2 + -3x + 5 = 0 | 1. Identify coefficients: a=9, b=-3, c=5.
2. Calculate the discriminant: Δ = b^2 - 4ac = -3^2 - 4(9)(5) = 9 - 180 = -171.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (3 ± √171i) / 18.
Final Answer: x = 0.16666666666666666 ± 0.7264831572567789i |
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... |
In a right-angled triangle, the lengths of the two legs are a = 4 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: 4^2 + 15^2 = c^2
2. Calculate the squares: 16 + 225 = c^2
3. Sum the squares: 241 = c^2
4. Take the square root of both sides: c = √241
Final Ans... |
Multiply the following 2x2 matrices:
A =
[-2, -1]
[0, -4]
B =
[-1, -5]
[0, -4] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-2*-1) + (-1*0) = 2
C12 = (A11 * B12) + (A12 * B22) = (-2*-5) + (-1*-4) = 14
C21 = (A21 * B11) + (A22 * B21) = (0*-1) + (-4*0) = 0
C22 = (A21 * B12) + (A22 * B22) = (0*-5) + (-4*-4) = 16
Res... |
Find the derivative of f(x) = (2x + 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 = 2x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 2x + 4. Its derivative is h'(x) = 2.
4. Multiply g'(u) by h'(x): f'(x) = 3(2x + 4)^2 * 2 = 6(2x +... |
Calculate the mean, population variance, and standard deviation for the dataset: [2, 12, 16, 10, 1] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (2 + 12 + 16 + 10 + 1) / 5 = 41 / 5 = 8.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-6.2, 3.8, 7.8, 1.8, -7.2]
Squared Differences: [38.44, 14.44, 60.84, 3.24, 51.84]
... |
In a right-angled triangle, the lengths of the two legs are a = 5 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: 5^2 + 4^2 = c^2
2. Calculate the squares: 25 + 16 = c^2
3. Sum the squares: 41 = c^2
4. Take the square root of both sides: c = √41
Final Answer:... |
Find the roots of the quadratic equation: 10x^2 + 9x + 3 = 0 | 1. Identify coefficients: a=10, b=9, c=3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 9^2 - 4(10)(3) = 81 - 120 = -39.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-9 ± √39i) / 20.
Final Answer: x = -0.45 ± 0.3122498999199199i |
Multiply the following 2x2 matrices:
A =
[-5, 0]
[-1, 2]
B =
[-3, -4]
[2, 5] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-5*-3) + (0*2) = 15
C12 = (A11 * B12) + (A12 * B22) = (-5*-4) + (0*5) = 20
C21 = (A21 * B11) + (A22 * B21) = (-1*-3) + (2*2) = 7
C22 = (A21 * B12) + (A22 * B22) = (-1*-4) + (2*5) = 14
Result... |
Add the fractions 5/3 and 6/12. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 3 and 12, which is 12.
2. Convert 5/3 by multiplying top and bottom by 4: 20/12.
3. Convert 6/12 by multiplying top and bottom by 1: 6/12.
4. Add the numerators: 20 + 6 = 26.
5. The resulting fraction is 26/12.
6. Simplify by div... |
Evaluate cos(90°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(90°) 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) |
Multiply the following 2x2 matrices:
A =
[-3, 0]
[0, -2]
B =
[1, 4]
[-2, 5] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-3*1) + (0*-2) = -3
C12 = (A11 * B12) + (A12 * B22) = (-3*4) + (0*5) = -12
C21 = (A21 * B11) + (A22 * B21) = (0*1) + (-2*-2) = 4
C22 = (A21 * B12) + (A22 * B22) = (0*4) + (-2*5) = -10
Result... |
Compute the indefinite integral of f(x) = 1x^2. | To find the indefinite integral ∫ 1x^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: 1 / 3 = 1/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 1x^2 dx = 1/3x^3 + C |
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) |
Multiply the following 2x2 matrices:
A =
[5, -3]
[1, 4]
B =
[5, 2]
[2, -2] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (5*5) + (-3*2) = 19
C12 = (A11 * B12) + (A12 * B22) = (5*2) + (-3*-2) = 16
C21 = (A21 * B11) + (A22 * B21) = (1*5) + (4*2) = 13
C22 = (A21 * B12) + (A22 * B22) = (1*2) + (4*-2) = -6
Result Ma... |
Add the fractions 9/14 and 8/6. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 14 and 6, which is 42.
2. Convert 9/14 by multiplying top and bottom by 3: 27/42.
3. Convert 8/6 by multiplying top and bottom by 7: 56/42.
4. Add the numerators: 27 + 56 = 83.
5. The resulting fraction is 83/42.
Final Answer: 83... |
Compute the quotient and remainder of 64359 divided by 51. Show the long division steps. | To divide 64359 by 51:
1. Determine how many times 51 goes into the leading digits of 64359.
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: 64359 / 51 = 1261 with a remainder of 48.
Proof: (1261 * 51... |
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) = (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 quotient and remainder of 33775 divided by 27. Show the long division steps. | To divide 33775 by 27:
1. Determine how many times 27 goes into the leading digits of 33775.
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: 33775 / 27 = 1250 with a remainder of 25.
Proof: (1250 * 27... |
Compute the quotient and remainder of 39474 divided by 32. Show the long division steps. | To divide 39474 by 32:
1. Determine how many times 32 goes into the leading digits of 39474.
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: 39474 / 32 = 1233 with a remainder of 18.
Proof: (1233 * 32... |
Find the roots of the quadratic equation: 8x^2 + 1x + -4 = 0 | 1. Identify coefficients: a=8, b=1, c=-4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1 - -128 = 129.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-1 ± √129) / 16.
5. Calculate roots: x1 = 0.6473635432250342, x2 = -0.7723635432250342.
Final ... |
Find the derivative of f(x) = (4x + 3)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 4x + 3.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 3. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 3)^2 * 4 = 12(4x ... |
Compute the indefinite integral of f(x) = 1x^1. | To find the indefinite integral ∫ 1x^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: 1 / 2 = 1/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 1x^1 dx = 1/2x^2 + C |
Find the roots of the quadratic equation: 4x^2 + 0x + 10 = 0 | 1. Identify coefficients: a=4, b=0, c=10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 0^2 - 4(4)(10) = 0 - 160 = -160.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (0 ± √160i) / 8.
Final Answer: x = 0.0 ± 1.5811388300841898i |
Evaluate the logarithm: log_4(64) | To evaluate log_4(64), we ask the question: '4 raised to what power equals 64?'
Let x be the unknown power: 4^x = 64
Since 4^3 = 64, it follows that x = 3.
Final Answer: 3 |
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) |
Find the roots of the quadratic equation: 8x^2 + -6x + -10 = 0 | 1. Identify coefficients: a=8, b=-6, c=-10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 36 - -320 = 356.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (6 ± √356) / 16.
5. Calculate roots: x1 = 1.5542476415070754, x2 = -0.8042476415070754.
Fina... |
Evaluate the logarithm: log_4(256) | To evaluate log_4(256), we ask the question: '4 raised to what power equals 256?'
Let x be the unknown power: 4^x = 256
Since 4^4 = 256, it follows that x = 4.
Final Answer: 4 |
Evaluate the logarithm: log_4(64) | To evaluate log_4(64), we ask the question: '4 raised to what power equals 64?'
Let x be the unknown power: 4^x = 64
Since 4^3 = 64, it follows that x = 3.
Final Answer: 3 |
Add the fractions 3/8 and 4/2. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 8 and 2, which is 8.
2. Convert 3/8 by multiplying top and bottom by 1: 3/8.
3. Convert 4/2 by multiplying top and bottom by 4: 16/8.
4. Add the numerators: 3 + 16 = 19.
5. The resulting fraction is 19/8.
Final Answer: 19/8 |
Add the fractions 4/15 and 11/2. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 15 and 2, which is 30.
2. Convert 4/15 by multiplying top and bottom by 2: 8/30.
3. Convert 11/2 by multiplying top and bottom by 15: 165/30.
4. Add the numerators: 8 + 165 = 173.
5. The resulting fraction is 173/30.
Final Answer... |
In a right-angled triangle, the lengths of the two legs are a = 4 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: 4^2 + 5^2 = c^2
2. Calculate the squares: 16 + 25 = c^2
3. Sum the squares: 41 = c^2
4. Take the square root of both sides: c = √41
Final Answer:... |
Multiply the following 2x2 matrices:
A =
[-5, 3]
[3, 0]
B =
[5, 0]
[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) = (-5*5) + (3*4) = -13
C12 = (A11 * B12) + (A12 * B22) = (-5*0) + (3*-4) = -12
C21 = (A21 * B11) + (A22 * B21) = (3*5) + (0*4) = 15
C22 = (A21 * B12) + (A22 * B22) = (3*0) + (0*-4) = 0
Result M... |
In a right-angled triangle, the lengths of the two legs are a = 8 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: 8^2 + 20^2 = c^2
2. Calculate the squares: 64 + 400 = c^2
3. Sum the squares: 464 = c^2
4. Take the square root of both sides: c = √464
Final Ans... |
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 |
Evaluate the logarithm: log_2(8) | To evaluate log_2(8), we ask the question: '2 raised to what power equals 8?'
Let x be the unknown power: 2^x = 8
Since 2^3 = 8, it follows that x = 3.
Final Answer: 3 |
Evaluate 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) |
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) |
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 |
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