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In a right-angled triangle, the lengths of the two legs are a = 7 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: 7^2 + 12^2 = c^2
2. Calculate the squares: 49 + 144 = c^2
3. Sum the squares: 193 = c^2
4. Take the square root of both sides: c = √193
Final Ans... |
Compute the quotient and remainder of 65854 divided by 17. Show the long division steps. | To divide 65854 by 17:
1. Determine how many times 17 goes into the leading digits of 65854.
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: 65854 / 17 = 3873 with a remainder of 13.
Proof: (3873 * 17... |
Multiply the following 2x2 matrices:
A =
[-1, -5]
[4, 3]
B =
[0, 5]
[2, 3] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-1*0) + (-5*2) = -10
C12 = (A11 * B12) + (A12 * B22) = (-1*5) + (-5*3) = -20
C21 = (A21 * B11) + (A22 * B21) = (4*0) + (3*2) = 6
C22 = (A21 * B12) + (A22 * B22) = (4*5) + (3*3) = 29
Result M... |
Calculate the mean, population variance, and standard deviation for the dataset: [2, 8, 16, 3, 18] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (2 + 8 + 16 + 3 + 18) / 5 = 47 / 5 = 9.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-7.4, -1.4, 6.6, -6.4, 8.6]
Squared Differences: [54.76, 1.96, 43.56, 40.96, 73.96]
... |
Add the fractions 12/11 and 4/5. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 11 and 5, which is 55.
2. Convert 12/11 by multiplying top and bottom by 5: 60/55.
3. Convert 4/5 by multiplying top and bottom by 11: 44/55.
4. Add the numerators: 60 + 44 = 104.
5. The resulting fraction is 104/55.
Final Answer... |
Compute the quotient and remainder of 91987 divided by 38. Show the long division steps. | To divide 91987 by 38:
1. Determine how many times 38 goes into the leading digits of 91987.
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: 91987 / 38 = 2420 with a remainder of 27.
Proof: (2420 * 38... |
Evaluate sin(0°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(0°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.0000 (exact form depends on root simplification if applicable) |
Calculate the mean, population variance, and standard deviation for the dataset: [16, 6, 15, 3, 11] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (16 + 6 + 15 + 3 + 11) / 5 = 51 / 5 = 10.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [5.8, -4.2, 4.8, -7.2, 0.8]
Squared Differences: [33.64, 17.64, 23.04, 51.84, 0.64]
... |
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) = 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(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 |
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 |
Compute the quotient and remainder of 56311 divided by 87. Show the long division steps. | To divide 56311 by 87:
1. Determine how many times 87 goes into the leading digits of 56311.
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: 56311 / 87 = 647 with a remainder of 22.
Proof: (647 * 87) ... |
Add the fractions 14/3 and 7/15. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 3 and 15, which is 15.
2. Convert 14/3 by multiplying top and bottom by 5: 70/15.
3. Convert 7/15 by multiplying top and bottom by 1: 7/15.
4. Add the numerators: 70 + 7 = 77.
5. The resulting fraction is 77/15.
Final Answer: 77/... |
Multiply the following 2x2 matrices:
A =
[-3, 4]
[0, 4]
B =
[-1, -1]
[-3, 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) + (4*-3) = -9
C12 = (A11 * B12) + (A12 * B22) = (-3*-1) + (4*5) = 23
C21 = (A21 * B11) + (A22 * B21) = (0*-1) + (4*-3) = -12
C22 = (A21 * B12) + (A22 * B22) = (0*-1) + (4*5) = 20
Resu... |
Calculate the mean, population variance, and standard deviation for the dataset: [1, 3, 17, 1, 4] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (1 + 3 + 17 + 1 + 4) / 5 = 26 / 5 = 5.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-4.2, -2.2, 11.8, -4.2, -1.2]
Squared Differences: [17.64, 4.84, 139.24, 17.64, 1.44]
... |
Compute the indefinite integral of f(x) = 9x^3. | To find the indefinite integral ∫ 9x^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: 9 / 4 = 9/4.
3. Combine and add the constant of integration C.
Final Answer: ∫ 9x^3 dx = 9/4x^4 + C |
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 |
Multiply the following 2x2 matrices:
A =
[-5, -2]
[5, -5]
B =
[-3, 5]
[1, -4] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-5*-3) + (-2*1) = 13
C12 = (A11 * B12) + (A12 * B22) = (-5*5) + (-2*-4) = -17
C21 = (A21 * B11) + (A22 * B21) = (5*-3) + (-5*1) = -20
C22 = (A21 * B12) + (A22 * B22) = (5*5) + (-5*-4) = 45
R... |
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 the logarithm: log_2(16) | To evaluate log_2(16), we ask the question: '2 raised to what power equals 16?'
Let x be the unknown power: 2^x = 16
Since 2^4 = 16, it follows that x = 4.
Final Answer: 4 |
Compute the indefinite integral of f(x) = 9x^2. | To find the indefinite integral ∫ 9x^2 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 2 + 1 = 3.
2. Divide the coefficient by the new exponent: 9 / 3 = 3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 9x^2 dx = 3x^3 + C |
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 ... |
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) |
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 |
Find the roots of the quadratic equation: 9x^2 + 7x + 0 = 0 | 1. Identify coefficients: a=9, 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) / 18.
5. Calculate roots: x1 = 0.0, x2 = -0.7777777777777778.
Final Answer: x = 0.0, -0.... |
Add the fractions 2/4 and 13/9. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 4 and 9, which is 36.
2. Convert 2/4 by multiplying top and bottom by 9: 18/36.
3. Convert 13/9 by multiplying top and bottom by 4: 52/36.
4. Add the numerators: 18 + 52 = 70.
5. The resulting fraction is 70/36.
6. Simplify by di... |
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) |
Evaluate the logarithm: log_5(125) | To evaluate log_5(125), we ask the question: '5 raised to what power equals 125?'
Let x be the unknown power: 5^x = 125
Since 5^3 = 125, it follows that x = 3.
Final Answer: 3 |
Compute the indefinite integral of f(x) = 6x^2. | To find the indefinite integral ∫ 6x^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: 6 / 3 = 2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 6x^2 dx = 2x^3 + 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 |
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... |
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 =
[-1, -1]
[-2, 1]
B =
[5, 1]
[-3, -4] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-1*5) + (-1*-3) = -2
C12 = (A11 * B12) + (A12 * B22) = (-1*1) + (-1*-4) = 3
C21 = (A21 * B11) + (A22 * B21) = (-2*5) + (1*-3) = -13
C22 = (A21 * B12) + (A22 * B22) = (-2*1) + (1*-4) = -6
Res... |
Multiply the following 2x2 matrices:
A =
[-2, -1]
[2, 5]
B =
[4, 3]
[-3, 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*4) + (-1*-3) = -5
C12 = (A11 * B12) + (A12 * B22) = (-2*3) + (-1*5) = -11
C21 = (A21 * B11) + (A22 * B21) = (2*4) + (5*-3) = -7
C22 = (A21 * B12) + (A22 * B22) = (2*3) + (5*5) = 31
Result... |
Multiply the following 2x2 matrices:
A =
[-4, 0]
[-1, 0]
B =
[0, -1]
[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) = (-4*0) + (0*1) = 0
C12 = (A11 * B12) + (A12 * B22) = (-4*-1) + (0*1) = 4
C21 = (A21 * B11) + (A22 * B21) = (-1*0) + (0*1) = 0
C22 = (A21 * B12) + (A22 * B22) = (-1*-1) + (0*1) = 1
Result Matr... |
Compute the quotient and remainder of 95157 divided by 90. Show the long division steps. | To divide 95157 by 90:
1. Determine how many times 90 goes into the leading digits of 95157.
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: 95157 / 90 = 1057 with a remainder of 27.
Proof: (1057 * 90... |
Calculate the mean, population variance, and standard deviation for the dataset: [19, 1, 20, 3, 14] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (19 + 1 + 20 + 3 + 14) / 5 = 57 / 5 = 11.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [7.6, -10.4, 8.6, -8.4, 2.6]
Squared Differences: [57.76, 108.16, 73.96, 70.56, 6.76... |
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 |
Add the fractions 7/9 and 2/14. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 9 and 14, which is 126.
2. Convert 7/9 by multiplying top and bottom by 14: 98/126.
3. Convert 2/14 by multiplying top and bottom by 9: 18/126.
4. Add the numerators: 98 + 18 = 116.
5. The resulting fraction is 116/126.
6. Simpli... |
Compute the indefinite integral of f(x) = 6x^4. | To find the indefinite integral ∫ 6x^4 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 4 + 1 = 5.
2. Divide the coefficient by the new exponent: 6 / 5 = 6/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 6x^4 dx = 6/5x^5 + C |
Compute the quotient and remainder of 50400 divided by 3. Show the long division steps. | To divide 50400 by 3:
1. Determine how many times 3 goes into the leading digits of 50400.
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: 50400 / 3 = 16800 with a remainder of 0.
Proof: (16800 * 3) +... |
Find the roots of the quadratic equation: 9x^2 + -8x + 10 = 0 | 1. Identify coefficients: a=9, b=-8, c=10.
2. Calculate the discriminant: Δ = b^2 - 4ac = -8^2 - 4(9)(10) = 64 - 360 = -296.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (8 ± √296i) / 18.
Final Answer: x = 0.4444444444444444 ± 0.9558139185602919i |
Find the roots of the quadratic equation: 4x^2 + 7x + 1 = 0 | 1. Identify coefficients: a=4, b=7, c=1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 49 - 16 = 33.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-7 ± √33) / 8.
5. Calculate roots: x1 = -0.15692966918274642, x2 = -1.5930703308172536.
Final Ans... |
Multiply the following 2x2 matrices:
A =
[-3, -1]
[-1, -3]
B =
[1, -1]
[-5, -1] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-3*1) + (-1*-5) = 2
C12 = (A11 * B12) + (A12 * B22) = (-3*-1) + (-1*-1) = 4
C21 = (A21 * B11) + (A22 * B21) = (-1*1) + (-3*-5) = 14
C22 = (A21 * B12) + (A22 * B22) = (-1*-1) + (-3*-1) = 4
Re... |
Calculate the mean, population variance, and standard deviation for the dataset: [12, 6, 8, 16, 20] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (12 + 6 + 8 + 16 + 20) / 5 = 62 / 5 = 12.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-0.4, -6.4, -4.4, 3.6, 7.6]
Squared Differences: [0.16, 40.96, 19.36, 12.96, 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 |
Calculate the mean, population variance, and standard deviation for the dataset: [14, 7, 11, 16, 10] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (14 + 7 + 11 + 16 + 10) / 5 = 58 / 5 = 11.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [2.4, -4.6, -0.6, 4.4, -1.6]
Squared Differences: [5.76, 21.16, 0.36, 19.36, 2.56]
... |
Evaluate the logarithm: log_3(81) | To evaluate log_3(81), we ask the question: '3 raised to what power equals 81?'
Let x be the unknown power: 3^x = 81
Since 3^4 = 81, it follows that x = 4.
Final Answer: 4 |
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 |
Add the fractions 6/3 and 13/6. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 3 and 6, which is 6.
2. Convert 6/3 by multiplying top and bottom by 2: 12/6.
3. Convert 13/6 by multiplying top and bottom by 1: 13/6.
4. Add the numerators: 12 + 13 = 25.
5. The resulting fraction is 25/6.
Final Answer: 25/6 |
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 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) |
Add the fractions 4/8 and 4/5. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 8 and 5, which is 40.
2. Convert 4/8 by multiplying top and bottom by 5: 20/40.
3. Convert 4/5 by multiplying top and bottom by 8: 32/40.
4. Add the numerators: 20 + 32 = 52.
5. The resulting fraction is 52/40.
6. Simplify by div... |
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 |
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 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) |
Compute the indefinite integral of f(x) = 9x^5. | To find the indefinite integral ∫ 9x^5 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 5 + 1 = 6.
2. Divide the coefficient by the new exponent: 9 / 6 = 3/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 9x^5 dx = 3/2x^6 + C |
Find the roots of the quadratic equation: 7x^2 + -3x + -7 = 0 | 1. Identify coefficients: a=7, b=-3, c=-7.
2. Calculate the discriminant: Δ = b^2 - 4ac = 9 - -196 = 205.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (3 ± √205) / 14.
5. Calculate roots: x1 = 1.236987218805454, x2 = -0.8084157902340252.
Final A... |
Multiply the following 2x2 matrices:
A =
[5, -4]
[0, -3]
B =
[-5, 5]
[-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*-5) + (-4*-2) = -17
C12 = (A11 * B12) + (A12 * B22) = (5*5) + (-4*-5) = 45
C21 = (A21 * B11) + (A22 * B21) = (0*-5) + (-3*-2) = 6
C22 = (A21 * B12) + (A22 * B22) = (0*5) + (-3*-5) = 15
Res... |
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 |
Compute the indefinite integral of f(x) = 5x^1. | To find the indefinite integral ∫ 5x^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: 5 / 2 = 5/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 5x^1 dx = 5/2x^2 + C |
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) |
Find the roots of the quadratic equation: 9x^2 + -1x + 0 = 0 | 1. Identify coefficients: a=9, b=-1, c=0.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1 - 0 = 1.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (1 ± √1) / 18.
5. Calculate roots: x1 = 0.1111111111111111, x2 = 0.0.
Final Answer: x = 0.1111111111... |
Find the roots of the quadratic equation: 1x^2 + 7x + 7 = 0 | 1. Identify coefficients: a=1, b=7, c=7.
2. Calculate the discriminant: Δ = b^2 - 4ac = 49 - 28 = 21.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-7 ± √21) / 2.
5. Calculate roots: x1 = -1.20871215252208, x2 = -5.7912878474779195.
Final Answer... |
Add the fractions 12/15 and 13/8. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 15 and 8, which is 120.
2. Convert 12/15 by multiplying top and bottom by 8: 96/120.
3. Convert 13/8 by multiplying top and bottom by 15: 195/120.
4. Add the numerators: 96 + 195 = 291.
5. The resulting fraction is 291/120.
6. Si... |
Find the roots of the quadratic equation: 3x^2 + -2x + -5 = 0 | 1. Identify coefficients: a=3, b=-2, c=-5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 4 - -60 = 64.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (2 ± √64) / 6.
5. Calculate roots: x1 = 1.6666666666666667, x2 = -1.0.
Final Answer: x = 1.66666... |
Add the fractions 11/14 and 11/8. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 14 and 8, which is 56.
2. Convert 11/14 by multiplying top and bottom by 4: 44/56.
3. Convert 11/8 by multiplying top and bottom by 7: 77/56.
4. Add the numerators: 44 + 77 = 121.
5. The resulting fraction is 121/56.
Final Answer... |
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 the logarithm: log_3(81) | To evaluate log_3(81), we ask the question: '3 raised to what power equals 81?'
Let x be the unknown power: 3^x = 81
Since 3^4 = 81, it follows that x = 4.
Final Answer: 4 |
Calculate the mean, population variance, and standard deviation for the dataset: [8, 9, 6, 5, 5] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (8 + 9 + 6 + 5 + 5) / 5 = 33 / 5 = 6.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [1.4, 2.4, -0.6, -1.6, -1.6]
Squared Differences: [1.96, 5.76, 0.36, 2.56, 2.56]
Sum ... |
Add the fractions 8/8 and 4/7. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 8 and 7, which is 56.
2. Convert 8/8 by multiplying top and bottom by 7: 56/56.
3. Convert 4/7 by multiplying top and bottom by 8: 32/56.
4. Add the numerators: 56 + 32 = 88.
5. The resulting fraction is 88/56.
6. Simplify by div... |
Multiply the following 2x2 matrices:
A =
[0, 5]
[-3, -2]
B =
[-2, 2]
[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) = (0*-2) + (5*4) = 20
C12 = (A11 * B12) + (A12 * B22) = (0*2) + (5*-5) = -25
C21 = (A21 * B11) + (A22 * B21) = (-3*-2) + (-2*4) = -2
C22 = (A21 * B12) + (A22 * B22) = (-3*2) + (-2*-5) = 4
Resul... |
Find the roots of the quadratic equation: 2x^2 + -10x + -6 = 0 | 1. Identify coefficients: a=2, b=-10, c=-6.
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) / 4.
5. Calculate roots: x1 = 5.541381265149109, x2 = -0.5413812651491097.
Final... |
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 |
Add the fractions 15/14 and 12/10. 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 15/14 by multiplying top and bottom by 5: 75/70.
3. Convert 12/10 by multiplying top and bottom by 7: 84/70.
4. Add the numerators: 75 + 84 = 159.
5. The resulting fraction is 159/70.
Final Answ... |
Multiply the following 2x2 matrices:
A =
[0, 0]
[-5, 4]
B =
[-1, -4]
[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) = (0*-1) + (0*1) = 0
C12 = (A11 * B12) + (A12 * B22) = (0*-4) + (0*-3) = 0
C21 = (A21 * B11) + (A22 * B21) = (-5*-1) + (4*1) = 9
C22 = (A21 * B12) + (A22 * B22) = (-5*-4) + (4*-3) = 8
Result Ma... |
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 = 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... |
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 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 +... |
Multiply the following 2x2 matrices:
A =
[2, 0]
[0, -1]
B =
[-2, -4]
[-5, -3] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (2*-2) + (0*-5) = -4
C12 = (A11 * B12) + (A12 * B22) = (2*-4) + (0*-3) = -8
C21 = (A21 * B11) + (A22 * B21) = (0*-2) + (-1*-5) = 5
C22 = (A21 * B12) + (A22 * B22) = (0*-4) + (-1*-3) = 3
Resul... |
Find the roots of the quadratic equation: 8x^2 + 9x + -7 = 0 | 1. Identify coefficients: a=8, b=9, c=-7.
2. Calculate the discriminant: Δ = b^2 - 4ac = 81 - -224 = 305.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-9 ± √305) / 16.
5. Calculate roots: x1 = 0.5290155747858112, x2 = -1.6540155747858112.
Final... |
Compute the quotient and remainder of 8831 divided by 22. Show the long division steps. | To divide 8831 by 22:
1. Determine how many times 22 goes into the leading digits of 8831.
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: 8831 / 22 = 401 with a remainder of 9.
Proof: (401 * 22) + 9 ... |
Calculate the mean, population variance, and standard deviation for the dataset: [2, 17, 8, 17, 5] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (2 + 17 + 8 + 17 + 5) / 5 = 49 / 5 = 9.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-7.8, 7.2, -1.8, 7.2, -4.8]
Squared Differences: [60.84, 51.84, 3.24, 51.84, 23.04]
... |
Compute the quotient and remainder of 59451 divided by 32. Show the long division steps. | To divide 59451 by 32:
1. Determine how many times 32 goes into the leading digits of 59451.
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: 59451 / 32 = 1857 with a remainder of 27.
Proof: (1857 * 32... |
Compute the quotient and remainder of 23844 divided by 34. Show the long division steps. | To divide 23844 by 34:
1. Determine how many times 34 goes into the leading digits of 23844.
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: 23844 / 34 = 701 with a remainder of 10.
Proof: (701 * 34) ... |
Add the fractions 3/13 and 3/6. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 13 and 6, which is 78.
2. Convert 3/13 by multiplying top and bottom by 6: 18/78.
3. Convert 3/6 by multiplying top and bottom by 13: 39/78.
4. Add the numerators: 18 + 39 = 57.
5. The resulting fraction is 57/78.
6. Simplify by ... |
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) |
Compute the quotient and remainder of 81573 divided by 95. Show the long division steps. | To divide 81573 by 95:
1. Determine how many times 95 goes into the leading digits of 81573.
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: 81573 / 95 = 858 with a remainder of 63.
Proof: (858 * 95) ... |
Compute the indefinite integral of f(x) = 2x^1. | To find the indefinite integral ∫ 2x^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: 2 / 2 = 1.
3. Combine and add the constant of integration C.
Final Answer: ∫ 2x^1 dx = 1x^2 + C |
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 = 6 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: 6^2 + 7^2 = c^2
2. Calculate the squares: 36 + 49 = c^2
3. Sum the squares: 85 = c^2
4. Take the square root of both sides: c = √85
Final Answer:... |
Compute the quotient and remainder of 37477 divided by 82. Show the long division steps. | To divide 37477 by 82:
1. Determine how many times 82 goes into the leading digits of 37477.
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: 37477 / 82 = 457 with a remainder of 3.
Proof: (457 * 82) +... |
Compute the indefinite integral of f(x) = 4x^3. | To find the indefinite integral ∫ 4x^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: 4 / 4 = 1.
3. Combine and add the constant of integration C.
Final Answer: ∫ 4x^3 dx = 1x^4 + C |
Evaluate cos(0°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(0°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 1.0000 (exact form depends on root simplification if applicable) |
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 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) = 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) = 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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