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In a right-angled triangle, the lengths of the two legs are a = 19 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: 19^2 + 12^2 = c^2
2. Calculate the squares: 361 + 144 = c^2
3. Sum the squares: 505 = c^2
4. Take the square root of both sides: c = √505
Final A... |
Calculate the mean, population variance, and standard deviation for the dataset: [12, 12, 19, 16, 19] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (12 + 12 + 19 + 16 + 19) / 5 = 78 / 5 = 15.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-3.6, -3.6, 3.4, 0.4, 3.4]
Squared Differences: [12.96, 12.96, 11.56, 0.16, 11.56... |
In a right-angled triangle, the lengths of the two legs are a = 12 and b = 6. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 12^2 + 6^2 = c^2
2. Calculate the squares: 144 + 36 = c^2
3. Sum the squares: 180 = c^2
4. Take the square root of both sides: c = √180
Final Ans... |
Find the roots of the quadratic equation: 7x^2 + -3x + -10 = 0 | 1. Identify coefficients: a=7, b=-3, c=-10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 9 - -280 = 289.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (3 ± √289) / 14.
5. Calculate roots: x1 = 1.4285714285714286, x2 = -1.0.
Final Answer: x = 1.... |
Compute the indefinite integral of f(x) = 6x^5. | To find the indefinite integral ∫ 6x^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: 6 / 6 = 1.
3. Combine and add the constant of integration C.
Final Answer: ∫ 6x^5 dx = 1x^6 + C |
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 |
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 |
Add the fractions 3/10 and 14/6. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 10 and 6, which is 30.
2. Convert 3/10 by multiplying top and bottom by 3: 9/30.
3. Convert 14/6 by multiplying top and bottom by 5: 70/30.
4. Add the numerators: 9 + 70 = 79.
5. The resulting fraction is 79/30.
Final Answer: 79/... |
Add the fractions 1/9 and 15/6. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 9 and 6, which is 18.
2. Convert 1/9 by multiplying top and bottom by 2: 2/18.
3. Convert 15/6 by multiplying top and bottom by 3: 45/18.
4. Add the numerators: 2 + 45 = 47.
5. The resulting fraction is 47/18.
Final Answer: 47/18 |
Compute the quotient and remainder of 40443 divided by 68. Show the long division steps. | To divide 40443 by 68:
1. Determine how many times 68 goes into the leading digits of 40443.
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: 40443 / 68 = 594 with a remainder of 51.
Proof: (594 * 68) ... |
Add the fractions 3/4 and 12/2. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 4 and 2, which is 4.
2. Convert 3/4 by multiplying top and bottom by 1: 3/4.
3. Convert 12/2 by multiplying top and bottom by 2: 24/4.
4. Add the numerators: 3 + 24 = 27.
5. The resulting fraction is 27/4.
Final Answer: 27/4 |
Evaluate sin(45°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(45°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.7071 (exact form depends on root simplification if applicable) |
Find the 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 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) = 3x^3. | To find the indefinite integral ∫ 3x^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: 3 / 4 = 3/4.
3. Combine and add the constant of integration C.
Final Answer: ∫ 3x^3 dx = 3/4x^4 + C |
In a right-angled triangle, the lengths of the two legs are a = 19 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: 19^2 + 17^2 = c^2
2. Calculate the squares: 361 + 289 = c^2
3. Sum the squares: 650 = c^2
4. Take the square root of both sides: c = √650
Final A... |
In a right-angled triangle, the lengths of the two legs are a = 11 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: 11^2 + 4^2 = c^2
2. Calculate the squares: 121 + 16 = c^2
3. Sum the squares: 137 = c^2
4. Take the square root of both sides: c = √137
Final Ans... |
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) |
Compute the indefinite integral of f(x) = 8x^4. | To find the indefinite integral ∫ 8x^4 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 4 + 1 = 5.
2. Divide the coefficient by the new exponent: 8 / 5 = 8/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 8x^4 dx = 8/5x^5 + C |
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) = 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 |
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 +... |
In a right-angled triangle, the lengths of the two legs are a = 18 and b = 6. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 18^2 + 6^2 = c^2
2. Calculate the squares: 324 + 36 = c^2
3. Sum the squares: 360 = c^2
4. Take the square root of both sides: c = √360
Final Ans... |
Compute the quotient and remainder of 67385 divided by 88. Show the long division steps. | To divide 67385 by 88:
1. Determine how many times 88 goes into the leading digits of 67385.
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: 67385 / 88 = 765 with a remainder of 65.
Proof: (765 * 88) ... |
Compute the quotient and remainder of 19784 divided by 23. Show the long division steps. | To divide 19784 by 23:
1. Determine how many times 23 goes into the leading digits of 19784.
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: 19784 / 23 = 860 with a remainder of 4.
Proof: (860 * 23) +... |
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 =
[4, 5]
[-2, -2]
B =
[2, 2]
[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) = (4*2) + (5*3) = 23
C12 = (A11 * B12) + (A12 * B22) = (4*2) + (5*-5) = -17
C21 = (A21 * B11) + (A22 * B21) = (-2*2) + (-2*3) = -10
C22 = (A21 * B12) + (A22 * B22) = (-2*2) + (-2*-5) = 6
Result... |
Multiply the following 2x2 matrices:
A =
[2, 3]
[-4, 0]
B =
[3, 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) = (2*3) + (3*2) = 12
C12 = (A11 * B12) + (A12 * B22) = (2*3) + (3*-1) = 3
C21 = (A21 * B11) + (A22 * B21) = (-4*3) + (0*2) = -12
C22 = (A21 * B12) + (A22 * B22) = (-4*3) + (0*-1) = -12
Result M... |
Find the roots of the quadratic equation: 2x^2 + 2x + -4 = 0 | 1. Identify coefficients: a=2, b=2, c=-4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 4 - -32 = 36.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-2 ± √36) / 4.
5. Calculate roots: x1 = 1.0, x2 = -2.0.
Final Answer: x = 1.0, -2.0 |
Multiply the following 2x2 matrices:
A =
[5, -3]
[5, 5]
B =
[-2, 2]
[0, -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) = (5*-2) + (-3*0) = -10
C12 = (A11 * B12) + (A12 * B22) = (5*2) + (-3*-3) = 19
C21 = (A21 * B11) + (A22 * B21) = (5*-2) + (5*0) = -10
C22 = (A21 * B12) + (A22 * B22) = (5*2) + (5*-3) = -5
Resul... |
Find the roots of the quadratic equation: 6x^2 + 10x + 4 = 0 | 1. Identify coefficients: a=6, b=10, c=4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 100 - 96 = 4.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-10 ± √4) / 12.
5. Calculate roots: x1 = -0.6666666666666666, x2 = -1.0.
Final Answer: x = -0.66... |
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 |
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 |
Multiply the following 2x2 matrices:
A =
[-5, 4]
[-4, -2]
B =
[3, 0]
[3, -1] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-5*3) + (4*3) = -3
C12 = (A11 * B12) + (A12 * B22) = (-5*0) + (4*-1) = -4
C21 = (A21 * B11) + (A22 * B21) = (-4*3) + (-2*3) = -18
C22 = (A21 * B12) + (A22 * B22) = (-4*0) + (-2*-1) = 2
Resul... |
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 tan(30°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(30°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.5774 (exact form depends on root simplification if applicable) |
Find the roots of the quadratic equation: 9x^2 + 6x + -10 = 0 | 1. Identify coefficients: a=9, b=6, c=-10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 36 - -360 = 396.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-6 ± √396) / 18.
5. Calculate roots: x1 = 0.7722082634517999, x2 = -1.4388749301184667.
Fina... |
Compute the quotient and remainder of 45262 divided by 14. Show the long division steps. | To divide 45262 by 14:
1. Determine how many times 14 goes into the leading digits of 45262.
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: 45262 / 14 = 3233 with a remainder of 0.
Proof: (3233 * 14)... |
Add the fractions 7/7 and 7/13. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 7 and 13, which is 91.
2. Convert 7/7 by multiplying top and bottom by 13: 91/91.
3. Convert 7/13 by multiplying top and bottom by 7: 49/91.
4. Add the numerators: 91 + 49 = 140.
5. The resulting fraction is 140/91.
6. Simplify b... |
Compute the indefinite integral of f(x) = 7x^1. | To find the indefinite integral ∫ 7x^1 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 1 + 1 = 2.
2. Divide the coefficient by the new exponent: 7 / 2 = 7/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 7x^1 dx = 7/2x^2 + C |
Multiply the following 2x2 matrices:
A =
[-4, -2]
[0, 0]
B =
[0, 2]
[4, 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) = (-4*0) + (-2*4) = -8
C12 = (A11 * B12) + (A12 * B22) = (-4*2) + (-2*3) = -14
C21 = (A21 * B11) + (A22 * B21) = (0*0) + (0*4) = 0
C22 = (A21 * B12) + (A22 * B22) = (0*2) + (0*3) = 0
Result Mat... |
Find the roots of the quadratic equation: 2x^2 + -5x + 3 = 0 | 1. Identify coefficients: a=2, b=-5, c=3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 25 - 24 = 1.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (5 ± √1) / 4.
5. Calculate roots: x1 = 1.5, x2 = 1.0.
Final Answer: x = 1.5, 1.0 |
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 +... |
Add the fractions 3/3 and 10/2. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 3 and 2, which is 6.
2. Convert 3/3 by multiplying top and bottom by 2: 6/6.
3. Convert 10/2 by multiplying top and bottom by 3: 30/6.
4. Add the numerators: 6 + 30 = 36.
5. The resulting fraction is 36/6.
6. Simplify by dividing... |
Find the roots of the quadratic equation: 8x^2 + 5x + 3 = 0 | 1. Identify coefficients: a=8, b=5, c=3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 5^2 - 4(8)(3) = 25 - 96 = -71.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-5 ± √71i) / 16.
Final Answer: x = -0.3125 ± 0.5266343608235224i |
Calculate the mean, population variance, and standard deviation for the dataset: [14, 1, 18, 1, 10] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (14 + 1 + 18 + 1 + 10) / 5 = 44 / 5 = 8.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [5.2, -7.8, 9.2, -7.8, 1.2]
Squared Differences: [27.04, 60.84, 84.64, 60.84, 1.44]
... |
Compute the quotient and remainder of 20712 divided by 34. Show the long division steps. | To divide 20712 by 34:
1. Determine how many times 34 goes into the leading digits of 20712.
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: 20712 / 34 = 609 with a remainder of 6.
Proof: (609 * 34) +... |
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 |
Evaluate the logarithm: log_2(32) | To evaluate log_2(32), we ask the question: '2 raised to what power equals 32?'
Let x be the unknown power: 2^x = 32
Since 2^5 = 32, it follows that x = 5.
Final Answer: 5 |
Calculate the mean, population variance, and standard deviation for the dataset: [2, 1, 14, 16, 6] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (2 + 1 + 14 + 16 + 6) / 5 = 39 / 5 = 7.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-5.8, -6.8, 6.2, 8.2, -1.8]
Squared Differences: [33.64, 46.24, 38.44, 67.24, 3.24]
... |
Find the derivative of f(x) = (5x + 3)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 5x + 3.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 5x + 3. Its derivative is h'(x) = 5.
4. Multiply g'(u) by h'(x): f'(x) = 3(5x + 3)^2 * 5 = 15(5x ... |
Add the fractions 8/10 and 2/11. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 10 and 11, which is 110.
2. Convert 8/10 by multiplying top and bottom by 11: 88/110.
3. Convert 2/11 by multiplying top and bottom by 10: 20/110.
4. Add the numerators: 88 + 20 = 108.
5. The resulting fraction is 108/110.
6. Sim... |
In a right-angled triangle, the lengths of the two legs are a = 14 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: 14^2 + 4^2 = c^2
2. Calculate the squares: 196 + 16 = c^2
3. Sum the squares: 212 = c^2
4. Take the square root of both sides: c = √212
Final Ans... |
Find the roots of the quadratic equation: 2x^2 + -7x + -10 = 0 | 1. Identify coefficients: a=2, b=-7, c=-10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 49 - -80 = 129.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (7 ± √129) / 4.
5. Calculate roots: x1 = 4.589454172900137, x2 = -1.0894541729001368.
Final A... |
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 |
Calculate the mean, population variance, and standard deviation for the dataset: [14, 9, 11, 17, 13] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (14 + 9 + 11 + 17 + 13) / 5 = 64 / 5 = 12.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [1.2, -3.8, -1.8, 4.2, 0.2]
Squared Differences: [1.44, 14.44, 3.24, 17.64, 0.04]
... |
Compute the quotient and remainder of 8899 divided by 15. Show the long division steps. | To divide 8899 by 15:
1. Determine how many times 15 goes into the leading digits of 8899.
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: 8899 / 15 = 593 with a remainder of 4.
Proof: (593 * 15) + 4 ... |
Compute the quotient and remainder of 38447 divided by 23. Show the long division steps. | To divide 38447 by 23:
1. Determine how many times 23 goes into the leading digits of 38447.
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: 38447 / 23 = 1671 with a remainder of 14.
Proof: (1671 * 23... |
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 |
Multiply the following 2x2 matrices:
A =
[-4, 3]
[0, -4]
B =
[-5, 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*-5) + (3*1) = 23
C12 = (A11 * B12) + (A12 * B22) = (-4*1) + (3*-1) = -7
C21 = (A21 * B11) + (A22 * B21) = (0*-5) + (-4*1) = -4
C22 = (A21 * B12) + (A22 * B22) = (0*1) + (-4*-1) = 4
Result... |
Compute the indefinite integral of f(x) = 8x^1. | To find the indefinite integral ∫ 8x^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: 8 / 2 = 4.
3. Combine and add the constant of integration C.
Final Answer: ∫ 8x^1 dx = 4x^2 + C |
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 |
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 = 3^2 - 4(10)(10) = 9 - 400 = -391.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-3 ± √391i) / 20.
Final Answer: x = -0.15 ± 0.9886859966642595i |
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 48052 divided by 70. Show the long division steps. | To divide 48052 by 70:
1. Determine how many times 70 goes into the leading digits of 48052.
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: 48052 / 70 = 686 with a remainder of 32.
Proof: (686 * 70) ... |
Add the fractions 4/8 and 12/15. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 8 and 15, which is 120.
2. Convert 4/8 by multiplying top and bottom by 15: 60/120.
3. Convert 12/15 by multiplying top and bottom by 8: 96/120.
4. Add the numerators: 60 + 96 = 156.
5. The resulting fraction is 156/120.
6. Simpl... |
Add the fractions 14/10 and 14/14. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 10 and 14, which is 70.
2. Convert 14/10 by multiplying top and bottom by 7: 98/70.
3. Convert 14/14 by multiplying top and bottom by 5: 70/70.
4. Add the numerators: 98 + 70 = 168.
5. The resulting fraction is 168/70.
6. Simplif... |
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 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 +... |
Multiply the following 2x2 matrices:
A =
[4, 0]
[4, 1]
B =
[-4, -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) = (4*-4) + (0*-1) = -16
C12 = (A11 * B12) + (A12 * B22) = (4*-5) + (0*-4) = -20
C21 = (A21 * B11) + (A22 * B21) = (4*-4) + (1*-1) = -17
C22 = (A21 * B12) + (A22 * B22) = (4*-5) + (1*-4) = -24
R... |
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 derivative of f(x) = (5x + 4)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 5x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 5x + 4. Its derivative is h'(x) = 5.
4. Multiply g'(u) by h'(x): f'(x) = 3(5x + 4)^2 * 5 = 15(5x ... |
Find the roots of the quadratic equation: 3x^2 + 1x + -8 = 0 | 1. Identify coefficients: a=3, b=1, c=-8.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1 - -96 = 97.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-1 ± √97) / 6.
5. Calculate roots: x1 = 1.474809633632684, x2 = -1.8081429669660174.
Final Answe... |
Evaluate the logarithm: log_2(32) | To evaluate log_2(32), we ask the question: '2 raised to what power equals 32?'
Let x be the unknown power: 2^x = 32
Since 2^5 = 32, it follows that x = 5.
Final Answer: 5 |
In a right-angled triangle, the lengths of the two legs are a = 4 and b = 8. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 4^2 + 8^2 = c^2
2. Calculate the squares: 16 + 64 = c^2
3. Sum the squares: 80 = c^2
4. Take the square root of both sides: c = √80
Final Answer:... |
Add the fractions 8/12 and 9/9. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 12 and 9, which is 36.
2. Convert 8/12 by multiplying top and bottom by 3: 24/36.
3. Convert 9/9 by multiplying top and bottom by 4: 36/36.
4. Add the numerators: 24 + 36 = 60.
5. The resulting fraction is 60/36.
6. Simplify by d... |
Evaluate tan(60°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(60°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 1.7321 (exact form depends on root simplification if applicable) |
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) = 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 |
Calculate the mean, population variance, and standard deviation for the dataset: [7, 4, 20, 6, 11] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (7 + 4 + 20 + 6 + 11) / 5 = 48 / 5 = 9.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-2.6, -5.6, 10.4, -3.6, 1.4]
Squared Differences: [6.76, 31.36, 108.16, 12.96, 1.96]
... |
In a right-angled triangle, the lengths of the two legs are a = 12 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: 12^2 + 3^2 = c^2
2. Calculate the squares: 144 + 9 = c^2
3. Sum the squares: 153 = c^2
4. Take the square root of both sides: c = √153
Final Answ... |
Evaluate the logarithm: log_2(8) | To evaluate log_2(8), we ask the question: '2 raised to what power equals 8?'
Let x be the unknown power: 2^x = 8
Since 2^3 = 8, it follows that x = 3.
Final Answer: 3 |
Find the roots of the quadratic equation: 9x^2 + -6x + -4 = 0 | 1. Identify coefficients: a=9, b=-6, c=-4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 36 - -144 = 180.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (6 ± √180) / 18.
5. Calculate roots: x1 = 1.0786893258332633, x2 = -0.4120226591665966.
Final... |
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 |
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 |
Find the roots of the quadratic equation: 7x^2 + 7x + 10 = 0 | 1. Identify coefficients: a=7, b=7, c=10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 7^2 - 4(7)(10) = 49 - 280 = -231.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-7 ± √231i) / 14.
Final Answer: x = -0.5 ± 1.0856202966836188i |
Multiply the following 2x2 matrices:
A =
[-1, 1]
[-3, -5]
B =
[3, 5]
[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*3) + (1*3) = 0
C12 = (A11 * B12) + (A12 * B22) = (-1*5) + (1*-4) = -9
C21 = (A21 * B11) + (A22 * B21) = (-3*3) + (-5*3) = -24
C22 = (A21 * B12) + (A22 * B22) = (-3*5) + (-5*-4) = 5
Result... |
Compute the quotient and remainder of 44710 divided by 62. Show the long division steps. | To divide 44710 by 62:
1. Determine how many times 62 goes into the leading digits of 44710.
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: 44710 / 62 = 721 with a remainder of 8.
Proof: (721 * 62) +... |
Multiply the following 2x2 matrices:
A =
[-5, -3]
[5, -4]
B =
[1, 1]
[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) = (-5*1) + (-3*5) = -20
C12 = (A11 * B12) + (A12 * B22) = (-5*1) + (-3*-2) = 1
C21 = (A21 * B11) + (A22 * B21) = (5*1) + (-4*5) = -15
C22 = (A21 * B12) + (A22 * B22) = (5*1) + (-4*-2) = 13
Resu... |
Find the roots of the quadratic equation: 4x^2 + 10x + -6 = 0 | 1. Identify coefficients: a=4, b=10, c=-6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 100 - -96 = 196.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-10 ± √196) / 8.
5. Calculate roots: x1 = 0.5, x2 = -3.0.
Final Answer: x = 0.5, -3.0 |
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... |
Compute the quotient and remainder of 11952 divided by 10. Show the long division steps. | To divide 11952 by 10:
1. Determine how many times 10 goes into the leading digits of 11952.
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: 11952 / 10 = 1195 with a remainder of 2.
Proof: (1195 * 10)... |
In a right-angled triangle, the lengths of the two legs are a = 8 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: 8^2 + 5^2 = c^2
2. Calculate the squares: 64 + 25 = c^2
3. Sum the squares: 89 = c^2
4. Take the square root of both sides: c = √89
Final Answer:... |
In a right-angled triangle, the lengths of the two legs are a = 16 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: 16^2 + 3^2 = c^2
2. Calculate the squares: 256 + 9 = c^2
3. Sum the squares: 265 = c^2
4. Take the square root of both sides: c = √265
Final Answ... |
Find the roots of the quadratic equation: 6x^2 + -1x + -6 = 0 | 1. Identify coefficients: a=6, b=-1, c=-6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1 - -144 = 145.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (1 ± √145) / 12.
5. Calculate roots: x1 = 1.0867995482326913, x2 = -0.9201328815660247.
Final ... |
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) = (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 ... |
Calculate the mean, population variance, and standard deviation for the dataset: [17, 14, 19, 9, 1] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (17 + 14 + 19 + 9 + 1) / 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, 2.0, 7.0, -3.0, -11.0]
Squared Differences: [25.0, 4.0, 49.0, 9.0, 121.0]
S... |
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 = 7 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: 7^2 + 16^2 = c^2
2. Calculate the squares: 49 + 256 = c^2
3. Sum the squares: 305 = c^2
4. Take the square root of both sides: c = √305
Final Ans... |
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