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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) = 10x^2. | To find the indefinite integral ∫ 10x^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: 10 / 3 = 10/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 10x^2 dx = 10/3x^3 + C |
Add the fractions 4/13 and 7/10. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 13 and 10, which is 130.
2. Convert 4/13 by multiplying top and bottom by 10: 40/130.
3. Convert 7/10 by multiplying top and bottom by 13: 91/130.
4. Add the numerators: 40 + 91 = 131.
5. The resulting fraction is 131/130.
Final ... |
Compute the indefinite integral of f(x) = 5x^3. | To find the indefinite integral ∫ 5x^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: 5 / 4 = 5/4.
3. Combine and add the constant of integration C.
Final Answer: ∫ 5x^3 dx = 5/4x^4 + C |
Multiply the following 2x2 matrices:
A =
[-4, 5]
[-3, -4]
B =
[-4, -3]
[-4, -1] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-4*-4) + (5*-4) = -4
C12 = (A11 * B12) + (A12 * B22) = (-4*-3) + (5*-1) = 7
C21 = (A21 * B11) + (A22 * B21) = (-3*-4) + (-4*-4) = 28
C22 = (A21 * B12) + (A22 * B22) = (-3*-3) + (-4*-1) = 13
... |
Multiply the following 2x2 matrices:
A =
[5, -2]
[0, 4]
B =
[0, 4]
[-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) = (5*0) + (-2*-4) = 8
C12 = (A11 * B12) + (A12 * B22) = (5*4) + (-2*-5) = 30
C21 = (A21 * B11) + (A22 * B21) = (0*0) + (4*-4) = -16
C22 = (A21 * B12) + (A22 * B22) = (0*4) + (4*-5) = -20
Result... |
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: 10x^2 + 3x + -10 = 0 | 1. Identify coefficients: a=10, b=3, c=-10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 9 - -400 = 409.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-3 ± √409) / 20.
5. Calculate roots: x1 = 0.8611874208078343, x2 = -1.1611874208078343.
Fina... |
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 67410 divided by 78. Show the long division steps. | To divide 67410 by 78:
1. Determine how many times 78 goes into the leading digits of 67410.
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: 67410 / 78 = 864 with a remainder of 18.
Proof: (864 * 78) ... |
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 |
Calculate the mean, population variance, and standard deviation for the dataset: [3, 8, 9, 20, 18] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (3 + 8 + 9 + 20 + 18) / 5 = 58 / 5 = 11.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-8.6, -3.6, -2.6, 8.4, 6.4]
Squared Differences: [73.96, 12.96, 6.76, 70.56, 40.96]
... |
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 quotient and remainder of 95763 divided by 72. Show the long division steps. | To divide 95763 by 72:
1. Determine how many times 72 goes into the leading digits of 95763.
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: 95763 / 72 = 1330 with a remainder of 3.
Proof: (1330 * 72)... |
Find the derivative of f(x) = (3x + 2)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 3x + 2.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 2. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 2)^2 * 3 = 9(3x +... |
In a right-angled triangle, the lengths of the two legs are a = 3 and b = 11. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 3^2 + 11^2 = c^2
2. Calculate the squares: 9 + 121 = c^2
3. Sum the squares: 130 = c^2
4. Take the square root of both sides: c = √130
Final Answ... |
In a right-angled triangle, the lengths of the two legs are a = 10 and b = 3. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 10^2 + 3^2 = c^2
2. Calculate the squares: 100 + 9 = c^2
3. Sum the squares: 109 = c^2
4. Take the square root of both sides: c = √109
Final Answ... |
Add the fractions 1/12 and 11/8. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 12 and 8, which is 24.
2. Convert 1/12 by multiplying top and bottom by 2: 2/24.
3. Convert 11/8 by multiplying top and bottom by 3: 33/24.
4. Add the numerators: 2 + 33 = 35.
5. The resulting fraction is 35/24.
Final Answer: 35/... |
Multiply the following 2x2 matrices:
A =
[-1, 1]
[4, 3]
B =
[-1, -4]
[-1, 4] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-1*-1) + (1*-1) = 0
C12 = (A11 * B12) + (A12 * B22) = (-1*-4) + (1*4) = 8
C21 = (A21 * B11) + (A22 * B21) = (4*-1) + (3*-1) = -7
C22 = (A21 * B12) + (A22 * B22) = (4*-4) + (3*4) = -4
Result ... |
Find the roots of the quadratic equation: 2x^2 + 1x + 6 = 0 | 1. Identify coefficients: a=2, b=1, c=6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1^2 - 4(2)(6) = 1 - 48 = -47.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-1 ± √47i) / 4.
Final Answer: x = -0.25 ± 1.713913650100261i |
Calculate the mean, population variance, and standard deviation for the dataset: [17, 2, 12, 15, 14] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (17 + 2 + 12 + 15 + 14) / 5 = 60 / 5 = 12.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [5.0, -10.0, 0.0, 3.0, 2.0]
Squared Differences: [25.0, 100.0, 0.0, 9.0, 4.0]
Su... |
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 |
Find the roots of the quadratic equation: 7x^2 + -10x + -7 = 0 | 1. Identify coefficients: a=7, b=-10, c=-7.
2. Calculate the discriminant: Δ = b^2 - 4ac = 100 - -196 = 296.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (10 ± √296) / 14.
5. Calculate roots: x1 = 1.9431893238632323, x2 = -0.5146178952918038.
Fi... |
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_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 |
Find the roots of the quadratic equation: 6x^2 + 10x + -2 = 0 | 1. Identify coefficients: a=6, b=10, c=-2.
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) / 12.
5. Calculate roots: x1 = 0.1804604217163699, x2 = -1.8471270883830364.
Fin... |
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 +... |
Evaluate the logarithm: log_5(3125) | To evaluate log_5(3125), we ask the question: '5 raised to what power equals 3125?'
Let x be the unknown power: 5^x = 3125
Since 5^5 = 3125, it follows that x = 5.
Final Answer: 5 |
Find the derivative of f(x) = (2x + 3)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 2x + 3.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 2x + 3. Its derivative is h'(x) = 2.
4. Multiply g'(u) by h'(x): f'(x) = 3(2x + 3)^2 * 2 = 6(2x +... |
Compute the quotient and remainder of 54251 divided by 38. Show the long division steps. | To divide 54251 by 38:
1. Determine how many times 38 goes into the leading digits of 54251.
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: 54251 / 38 = 1427 with a remainder of 25.
Proof: (1427 * 38... |
Compute the quotient and remainder of 60243 divided by 88. Show the long division steps. | To divide 60243 by 88:
1. Determine how many times 88 goes into the leading digits of 60243.
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: 60243 / 88 = 684 with a remainder of 51.
Proof: (684 * 88) ... |
Find the roots of the quadratic equation: 4x^2 + 8x + 10 = 0 | 1. Identify coefficients: a=4, b=8, c=10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 8^2 - 4(4)(10) = 64 - 160 = -96.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-8 ± √96i) / 8.
Final Answer: x = -1.0 ± 1.224744871391589i |
Compute the quotient and remainder of 74767 divided by 97. Show the long division steps. | To divide 74767 by 97:
1. Determine how many times 97 goes into the leading digits of 74767.
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: 74767 / 97 = 770 with a remainder of 77.
Proof: (770 * 97) ... |
Add the fractions 4/6 and 7/3. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 6 and 3, which is 6.
2. Convert 4/6 by multiplying top and bottom by 1: 4/6.
3. Convert 7/3 by multiplying top and bottom by 2: 14/6.
4. Add the numerators: 4 + 14 = 18.
5. The resulting fraction is 18/6.
6. Simplify by dividing ... |
Compute the quotient and remainder of 71545 divided by 63. Show the long division steps. | To divide 71545 by 63:
1. Determine how many times 63 goes into the leading digits of 71545.
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: 71545 / 63 = 1135 with a remainder of 40.
Proof: (1135 * 63... |
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 the logarithm: log_4(1024) | To evaluate log_4(1024), we ask the question: '4 raised to what power equals 1024?'
Let x be the unknown power: 4^x = 1024
Since 4^5 = 1024, it follows that x = 5.
Final Answer: 5 |
Evaluate 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 = 18 and b = 16. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 18^2 + 16^2 = c^2
2. Calculate the squares: 324 + 256 = c^2
3. Sum the squares: 580 = c^2
4. Take the square root of both sides: c = √580
Final A... |
Add the fractions 13/6 and 15/10. 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 13/6 by multiplying top and bottom by 5: 65/30.
3. Convert 15/10 by multiplying top and bottom by 3: 45/30.
4. Add the numerators: 65 + 45 = 110.
5. The resulting fraction is 110/30.
6. Simplify ... |
Calculate the mean, population variance, and standard deviation for the dataset: [19, 3, 7, 17, 20] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (19 + 3 + 7 + 17 + 20) / 5 = 66 / 5 = 13.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [5.8, -10.2, -6.2, 3.8, 6.8]
Squared Differences: [33.64, 104.04, 38.44, 14.44, 46.2... |
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) = 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 =
[1, 0]
[0, 2]
B =
[-5, 0]
[-1, 1] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (1*-5) + (0*-1) = -5
C12 = (A11 * B12) + (A12 * B22) = (1*0) + (0*1) = 0
C21 = (A21 * B11) + (A22 * B21) = (0*-5) + (2*-1) = -2
C22 = (A21 * B12) + (A22 * B22) = (0*0) + (2*1) = 2
Result Matr... |
Find the roots of the quadratic equation: 5x^2 + -9x + 9 = 0 | 1. Identify coefficients: a=5, b=-9, c=9.
2. Calculate the discriminant: Δ = b^2 - 4ac = -9^2 - 4(5)(9) = 81 - 180 = -99.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (9 ± √99i) / 10.
Final Answer: x = 0.9 ± 0.99498743710662i |
Calculate the mean, population variance, and standard deviation for the dataset: [16, 7, 13, 2, 14] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (16 + 7 + 13 + 2 + 14) / 5 = 52 / 5 = 10.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [5.6, -3.4, 2.6, -8.4, 3.6]
Squared Differences: [31.36, 11.56, 6.76, 70.56, 12.96]
... |
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 9/14 and 5/6. 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 5/6 by multiplying top and bottom by 7: 35/42.
4. Add the numerators: 27 + 35 = 62.
5. The resulting fraction is 62/42.
6. Simplify by d... |
Compute the quotient and remainder of 41638 divided by 91. Show the long division steps. | To divide 41638 by 91:
1. Determine how many times 91 goes into the leading digits of 41638.
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: 41638 / 91 = 457 with a remainder of 51.
Proof: (457 * 91) ... |
Compute the indefinite integral of f(x) = 2x^2. | To find the indefinite integral ∫ 2x^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: 2 / 3 = 2/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 2x^2 dx = 2/3x^3 + C |
Compute the indefinite integral of f(x) = 2x^3. | To find the indefinite integral ∫ 2x^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: 2 / 4 = 1/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 2x^3 dx = 1/2x^4 + 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 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 |
Compute the quotient and remainder of 51207 divided by 79. Show the long division steps. | To divide 51207 by 79:
1. Determine how many times 79 goes into the leading digits of 51207.
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: 51207 / 79 = 648 with a remainder of 15.
Proof: (648 * 79) ... |
Find the derivative of f(x) = (4x + 2)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 4x + 2.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 2. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 2)^2 * 4 = 12(4x ... |
Compute the quotient and remainder of 77371 divided by 61. Show the long division steps. | To divide 77371 by 61:
1. Determine how many times 61 goes into the leading digits of 77371.
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: 77371 / 61 = 1268 with a remainder of 23.
Proof: (1268 * 61... |
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) |
Multiply the following 2x2 matrices:
A =
[3, -5]
[-4, -4]
B =
[4, -5]
[-1, 2] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (3*4) + (-5*-1) = 17
C12 = (A11 * B12) + (A12 * B22) = (3*-5) + (-5*2) = -25
C21 = (A21 * B11) + (A22 * B21) = (-4*4) + (-4*-1) = -12
C22 = (A21 * B12) + (A22 * B22) = (-4*-5) + (-4*2) = 12
R... |
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 |
Compute the quotient and remainder of 36139 divided by 63. Show the long division steps. | To divide 36139 by 63:
1. Determine how many times 63 goes into the leading digits of 36139.
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: 36139 / 63 = 573 with a remainder of 40.
Proof: (573 * 63) ... |
Find the derivative of f(x) = (3x + 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 = 3x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 5. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 5)^2 * 3 = 9(3x +... |
Multiply the following 2x2 matrices:
A =
[-3, -1]
[4, -4]
B =
[-2, 4]
[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*-2) + (-1*5) = 1
C12 = (A11 * B12) + (A12 * B22) = (-3*4) + (-1*5) = -17
C21 = (A21 * B11) + (A22 * B21) = (4*-2) + (-4*5) = -28
C22 = (A21 * B12) + (A22 * B22) = (4*4) + (-4*5) = -4
Resu... |
Find the derivative of f(x) = (2x + 2)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 2x + 2.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 2x + 2. Its derivative is h'(x) = 2.
4. Multiply g'(u) by h'(x): f'(x) = 3(2x + 2)^2 * 2 = 6(2x +... |
Find the roots of the quadratic equation: 3x^2 + -3x + 8 = 0 | 1. Identify coefficients: a=3, b=-3, c=8.
2. Calculate the discriminant: Δ = b^2 - 4ac = -3^2 - 4(3)(8) = 9 - 96 = -87.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (3 ± √87i) / 6.
Final Answer: x = 0.5 ± 1.5545631755148026i |
Multiply the following 2x2 matrices:
A =
[0, 5]
[2, -4]
B =
[1, -2]
[1, -2] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (0*1) + (5*1) = 5
C12 = (A11 * B12) + (A12 * B22) = (0*-2) + (5*-2) = -10
C21 = (A21 * B11) + (A22 * B21) = (2*1) + (-4*1) = -2
C22 = (A21 * B12) + (A22 * B22) = (2*-2) + (-4*-2) = 4
Result M... |
Find the roots of the quadratic equation: 7x^2 + 6x + 4 = 0 | 1. Identify coefficients: a=7, b=6, c=4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 6^2 - 4(7)(4) = 36 - 112 = -76.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-6 ± √76i) / 14.
Final Answer: x = -0.42857142857142855 ± 0.6226998490772392i |
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) |
Compute the quotient and remainder of 33847 divided by 48. Show the long division steps. | To divide 33847 by 48:
1. Determine how many times 48 goes into the leading digits of 33847.
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: 33847 / 48 = 705 with a remainder of 7.
Proof: (705 * 48) +... |
Find the roots of the quadratic equation: 9x^2 + -3x + -10 = 0 | 1. Identify coefficients: a=9, b=-3, c=-10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 9 - -360 = 369.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (3 ± √369) / 18.
5. Calculate roots: x1 = 1.2338540395721413, x2 = -0.9005207062388081.
Final... |
In a right-angled triangle, the lengths of the two legs are a = 18 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: 18^2 + 18^2 = c^2
2. Calculate the squares: 324 + 324 = c^2
3. Sum the squares: 648 = c^2
4. Take the square root of both sides: c = √648
Final A... |
Find the roots of the quadratic equation: 2x^2 + 7x + -9 = 0 | 1. Identify coefficients: a=2, b=7, c=-9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 49 - -72 = 121.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-7 ± √121) / 4.
5. Calculate roots: x1 = 1.0, x2 = -4.5.
Final Answer: x = 1.0, -4.5 |
Compute the quotient and remainder of 50972 divided by 55. Show the long division steps. | To divide 50972 by 55:
1. Determine how many times 55 goes into the leading digits of 50972.
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: 50972 / 55 = 926 with a remainder of 42.
Proof: (926 * 55) ... |
Add the fractions 5/4 and 1/13. 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 5/4 by multiplying top and bottom by 13: 65/52.
3. Convert 1/13 by multiplying top and bottom by 4: 4/52.
4. Add the numerators: 65 + 4 = 69.
5. The resulting fraction is 69/52.
Final Answer: 69/... |
Multiply the following 2x2 matrices:
A =
[-2, 4]
[-1, 5]
B =
[-3, 4]
[-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*-3) + (4*-5) = -14
C12 = (A11 * B12) + (A12 * B22) = (-2*4) + (4*-5) = -28
C21 = (A21 * B11) + (A22 * B21) = (-1*-3) + (5*-5) = -22
C22 = (A21 * B12) + (A22 * B22) = (-1*4) + (5*-5) = -29
... |
Find the derivative of f(x) = (4x + 2)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 4x + 2.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 2. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 2)^2 * 4 = 12(4x ... |
Compute the quotient and remainder of 93527 divided by 40. Show the long division steps. | To divide 93527 by 40:
1. Determine how many times 40 goes into the leading digits of 93527.
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: 93527 / 40 = 2338 with a remainder of 7.
Proof: (2338 * 40)... |
Add the fractions 10/8 and 12/13. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 8 and 13, which is 104.
2. Convert 10/8 by multiplying top and bottom by 13: 130/104.
3. Convert 12/13 by multiplying top and bottom by 8: 96/104.
4. Add the numerators: 130 + 96 = 226.
5. The resulting fraction is 226/104.
6. Si... |
Evaluate the logarithm: log_3(27) | To evaluate log_3(27), we ask the question: '3 raised to what power equals 27?'
Let x be the unknown power: 3^x = 27
Since 3^3 = 27, it follows that x = 3.
Final Answer: 3 |
Find the derivative of f(x) = (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 quotient and remainder of 4383 divided by 10. Show the long division steps. | To divide 4383 by 10:
1. Determine how many times 10 goes into the leading digits of 4383.
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: 4383 / 10 = 438 with a remainder of 3.
Proof: (438 * 10) + 3 ... |
In a right-angled triangle, the lengths of the two legs are a = 9 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: 9^2 + 15^2 = c^2
2. Calculate the squares: 81 + 225 = c^2
3. Sum the squares: 306 = c^2
4. Take the square root of both sides: c = √306
Final Ans... |
Multiply the following 2x2 matrices:
A =
[-5, -3]
[-4, 2]
B =
[4, 1]
[-4, 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) = (-5*4) + (-3*-4) = -8
C12 = (A11 * B12) + (A12 * B22) = (-5*1) + (-3*0) = -5
C21 = (A21 * B11) + (A22 * B21) = (-4*4) + (2*-4) = -24
C22 = (A21 * B12) + (A22 * B22) = (-4*1) + (2*0) = -4
Resu... |
Multiply the following 2x2 matrices:
A =
[-2, 3]
[1, 5]
B =
[-3, 1]
[-5, 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) = (-2*-3) + (3*-5) = -9
C12 = (A11 * B12) + (A12 * B22) = (-2*1) + (3*0) = -2
C21 = (A21 * B11) + (A22 * B21) = (1*-3) + (5*-5) = -28
C22 = (A21 * B12) + (A22 * B22) = (1*1) + (5*0) = 1
Result ... |
Multiply the following 2x2 matrices:
A =
[-2, -4]
[1, -4]
B =
[5, 5]
[1, 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) + (-4*1) = -14
C12 = (A11 * B12) + (A12 * B22) = (-2*5) + (-4*5) = -30
C21 = (A21 * B11) + (A22 * B21) = (1*5) + (-4*1) = 1
C22 = (A21 * B12) + (A22 * B22) = (1*5) + (-4*5) = -15
Resul... |
Add the fractions 14/5 and 13/7. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 5 and 7, which is 35.
2. Convert 14/5 by multiplying top and bottom by 7: 98/35.
3. Convert 13/7 by multiplying top and bottom by 5: 65/35.
4. Add the numerators: 98 + 65 = 163.
5. The resulting fraction is 163/35.
Final Answer: ... |
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) |
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 derivative of f(x) = (3x + 4)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 3x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 4. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 4)^2 * 3 = 9(3x +... |
Evaluate tan(45°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(45°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 1.0000 (exact form depends on root simplification if applicable) |
Multiply the following 2x2 matrices:
A =
[1, 3]
[0, -1]
B =
[-5, -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) = (1*-5) + (3*-2) = -11
C12 = (A11 * B12) + (A12 * B22) = (1*-4) + (3*5) = 11
C21 = (A21 * B11) + (A22 * B21) = (0*-5) + (-1*-2) = 2
C22 = (A21 * B12) + (A22 * B22) = (0*-4) + (-1*5) = -5
Resul... |
Add the fractions 10/5 and 7/3. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 5 and 3, which is 15.
2. Convert 10/5 by multiplying top and bottom by 3: 30/15.
3. Convert 7/3 by multiplying top and bottom by 5: 35/15.
4. Add the numerators: 30 + 35 = 65.
5. The resulting fraction is 65/15.
6. Simplify by di... |
Compute the quotient and remainder of 11113 divided by 21. Show the long division steps. | To divide 11113 by 21:
1. Determine how many times 21 goes into the leading digits of 11113.
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: 11113 / 21 = 529 with a remainder of 4.
Proof: (529 * 21) +... |
Calculate the mean, population variance, and standard deviation for the dataset: [5, 14, 1, 3, 16] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (5 + 14 + 1 + 3 + 16) / 5 = 39 / 5 = 7.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-2.8, 6.2, -6.8, -4.8, 8.2]
Squared Differences: [7.84, 38.44, 46.24, 23.04, 67.24]
... |
Multiply the following 2x2 matrices:
A =
[3, -4]
[-5, -2]
B =
[-4, 1]
[-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) = (3*-4) + (-4*-1) = -8
C12 = (A11 * B12) + (A12 * B22) = (3*1) + (-4*4) = -13
C21 = (A21 * B11) + (A22 * B21) = (-5*-4) + (-2*-1) = 22
C22 = (A21 * B12) + (A22 * B22) = (-5*1) + (-2*4) = -13
R... |
Calculate the mean, population variance, and standard deviation for the dataset: [12, 12, 7, 6, 3] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (12 + 12 + 7 + 6 + 3) / 5 = 40 / 5 = 8.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [4.0, 4.0, -1.0, -2.0, -5.0]
Squared Differences: [16.0, 16.0, 1.0, 4.0, 25.0]
Sum ... |
Calculate the mean, population variance, and standard deviation for the dataset: [5, 12, 8, 4, 10] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (5 + 12 + 8 + 4 + 10) / 5 = 39 / 5 = 7.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-2.8, 4.2, 0.2, -3.8, 2.2]
Squared Differences: [7.84, 17.64, 0.04, 14.44, 4.84]
S... |
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 |
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 +... |
Multiply the following 2x2 matrices:
A =
[0, -1]
[2, -4]
B =
[-4, -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) = (0*-4) + (-1*1) = -1
C12 = (A11 * B12) + (A12 * B22) = (0*-1) + (-1*1) = -1
C21 = (A21 * B11) + (A22 * B21) = (2*-4) + (-4*1) = -12
C22 = (A21 * B12) + (A22 * B22) = (2*-1) + (-4*1) = -6
Resu... |
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... |
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