prompt stringlengths 32 115 | completion stringlengths 167 566 |
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Calculate the mean, population variance, and standard deviation for the dataset: [1, 20, 10, 9, 18] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (1 + 20 + 10 + 9 + 18) / 5 = 58 / 5 = 11.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-10.6, 8.4, -1.6, -2.6, 6.4]
Squared Differences: [112.36, 70.56, 2.56, 6.76, 40.96... |
Compute the quotient and remainder of 35182 divided by 75. Show the long division steps. | To divide 35182 by 75:
1. Determine how many times 75 goes into the leading digits of 35182.
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: 35182 / 75 = 469 with a remainder of 7.
Proof: (469 * 75) +... |
Add the fractions 12/5 and 4/4. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 5 and 4, which is 20.
2. Convert 12/5 by multiplying top and bottom by 4: 48/20.
3. Convert 4/4 by multiplying top and bottom by 5: 20/20.
4. Add the numerators: 48 + 20 = 68.
5. The resulting fraction is 68/20.
6. Simplify by di... |
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) |
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(25) | To evaluate log_5(25), we ask the question: '5 raised to what power equals 25?'
Let x be the unknown power: 5^x = 25
Since 5^2 = 25, it follows that x = 2.
Final Answer: 2 |
Find the roots of the quadratic equation: 6x^2 + -4x + 10 = 0 | 1. Identify coefficients: a=6, b=-4, c=10.
2. Calculate the discriminant: Δ = b^2 - 4ac = -4^2 - 4(6)(10) = 16 - 240 = -224.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (4 ± √224i) / 12.
Final Answer: x = 0.3333333333333333 ± 1.247219128924647i |
Multiply the following 2x2 matrices:
A =
[3, 5]
[3, -3]
B =
[3, -4]
[2, -1] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (3*3) + (5*2) = 19
C12 = (A11 * B12) + (A12 * B22) = (3*-4) + (5*-1) = -17
C21 = (A21 * B11) + (A22 * B21) = (3*3) + (-3*2) = 3
C22 = (A21 * B12) + (A22 * B22) = (3*-4) + (-3*-1) = -9
Result ... |
In a right-angled triangle, the lengths of the two legs are a = 13 and b = 13. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 13^2 + 13^2 = c^2
2. Calculate the squares: 169 + 169 = c^2
3. Sum the squares: 338 = c^2
4. Take the square root of both sides: c = √338
Final A... |
In a right-angled triangle, the lengths of the two legs are a = 15 and b = 10. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 15^2 + 10^2 = c^2
2. Calculate the squares: 225 + 100 = c^2
3. Sum the squares: 325 = c^2
4. Take the square root of both sides: c = √325
Final A... |
In a right-angled triangle, the lengths of the two legs are a = 16 and b = 14. 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 + 14^2 = c^2
2. Calculate the squares: 256 + 196 = c^2
3. Sum the squares: 452 = c^2
4. Take the square root of both sides: c = √452
Final A... |
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) = 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 |
In a right-angled triangle, the lengths of the two legs are a = 15 and b = 14. 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 + 14^2 = c^2
2. Calculate the squares: 225 + 196 = c^2
3. Sum the squares: 421 = c^2
4. Take the square root of both sides: c = √421
Final A... |
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 +... |
Find the roots of the quadratic equation: 5x^2 + -3x + 8 = 0 | 1. Identify coefficients: a=5, b=-3, c=8.
2. Calculate the discriminant: Δ = b^2 - 4ac = -3^2 - 4(5)(8) = 9 - 160 = -151.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (3 ± √151i) / 10.
Final Answer: x = 0.3 ± 1.2288205727444508i |
Add the fractions 7/10 and 9/2. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 10 and 2, which is 10.
2. Convert 7/10 by multiplying top and bottom by 1: 7/10.
3. Convert 9/2 by multiplying top and bottom by 5: 45/10.
4. Add the numerators: 7 + 45 = 52.
5. The resulting fraction is 52/10.
6. Simplify by div... |
Multiply the following 2x2 matrices:
A =
[3, -3]
[5, 5]
B =
[1, -4]
[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) + (-3*5) = -12
C12 = (A11 * B12) + (A12 * B22) = (3*-4) + (-3*1) = -15
C21 = (A21 * B11) + (A22 * B21) = (5*1) + (5*5) = 30
C22 = (A21 * B12) + (A22 * B22) = (5*-4) + (5*1) = -15
Result... |
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 |
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 |
Multiply the following 2x2 matrices:
A =
[-4, -2]
[1, 0]
B =
[1, 5]
[-3, 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*1) + (-2*-3) = 2
C12 = (A11 * B12) + (A12 * B22) = (-4*5) + (-2*3) = -26
C21 = (A21 * B11) + (A22 * B21) = (1*1) + (0*-3) = 1
C22 = (A21 * B12) + (A22 * B22) = (1*5) + (0*3) = 5
Result Ma... |
Multiply the following 2x2 matrices:
A =
[-2, 0]
[-3, 0]
B =
[4, 4]
[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) = (-2*4) + (0*4) = -8
C12 = (A11 * B12) + (A12 * B22) = (-2*4) + (0*-3) = -8
C21 = (A21 * B11) + (A22 * B21) = (-3*4) + (0*4) = -12
C22 = (A21 * B12) + (A22 * B22) = (-3*4) + (0*-3) = -12
Resul... |
Add the fractions 14/14 and 12/2. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 14 and 2, which is 14.
2. Convert 14/14 by multiplying top and bottom by 1: 14/14.
3. Convert 12/2 by multiplying top and bottom by 7: 84/14.
4. Add the numerators: 14 + 84 = 98.
5. The resulting fraction is 98/14.
6. Simplify by... |
Find the roots of the quadratic equation: 3x^2 + -8x + 7 = 0 | 1. Identify coefficients: a=3, b=-8, c=7.
2. Calculate the discriminant: Δ = b^2 - 4ac = -8^2 - 4(3)(7) = 64 - 84 = -20.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (8 ± √20i) / 6.
Final Answer: x = 1.3333333333333333 ± 0.7453559924999299i |
Compute the indefinite integral of f(x) = 7x^2. | To find the indefinite integral ∫ 7x^2 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 2 + 1 = 3.
2. Divide the coefficient by the new exponent: 7 / 3 = 7/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 7x^2 dx = 7/3x^3 + C |
Multiply the following 2x2 matrices:
A =
[-1, 4]
[3, 4]
B =
[0, -2]
[0, 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*0) + (4*0) = 0
C12 = (A11 * B12) + (A12 * B22) = (-1*-2) + (4*5) = 22
C21 = (A21 * B11) + (A22 * B21) = (3*0) + (4*0) = 0
C22 = (A21 * B12) + (A22 * B22) = (3*-2) + (4*5) = 14
Result Matr... |
In a right-angled triangle, the lengths of the two legs are a = 13 and b = 20. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 13^2 + 20^2 = c^2
2. Calculate the squares: 169 + 400 = c^2
3. Sum the squares: 569 = c^2
4. Take the square root of both sides: c = √569
Final A... |
Calculate the mean, population variance, and standard deviation for the dataset: [6, 9, 20, 19, 4] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (6 + 9 + 20 + 19 + 4) / 5 = 58 / 5 = 11.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-5.6, -2.6, 8.4, 7.4, -7.6]
Squared Differences: [31.36, 6.76, 70.56, 54.76, 57.76]
... |
Calculate the mean, population variance, and standard deviation for the dataset: [17, 12, 2, 16, 20] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (17 + 12 + 2 + 16 + 20) / 5 = 67 / 5 = 13.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [3.6, -1.4, -11.4, 2.6, 6.6]
Squared Differences: [12.96, 1.96, 129.96, 6.76, 43.56... |
Find the roots of the quadratic equation: 7x^2 + -8x + -1 = 0 | 1. Identify coefficients: a=7, b=-8, c=-1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 64 - -28 = 92.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (8 ± √92) / 14.
5. Calculate roots: x1 = 1.2565473604732456, x2 = -0.11369021761610273.
Final A... |
Multiply the following 2x2 matrices:
A =
[0, -4]
[-3, 4]
B =
[5, 0]
[-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) = (0*5) + (-4*-2) = 8
C12 = (A11 * B12) + (A12 * B22) = (0*0) + (-4*1) = -4
C21 = (A21 * B11) + (A22 * B21) = (-3*5) + (4*-2) = -23
C22 = (A21 * B12) + (A22 * B22) = (-3*0) + (4*1) = 4
Result M... |
Compute the indefinite integral of f(x) = 9x^4. | To find the indefinite integral ∫ 9x^4 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 4 + 1 = 5.
2. Divide the coefficient by the new exponent: 9 / 5 = 9/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 9x^4 dx = 9/5x^5 + C |
Evaluate the logarithm: log_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 |
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 quotient and remainder of 66513 divided by 98. Show the long division steps. | To divide 66513 by 98:
1. Determine how many times 98 goes into the leading digits of 66513.
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: 66513 / 98 = 678 with a remainder of 69.
Proof: (678 * 98) ... |
Add the fractions 11/10 and 13/13. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 10 and 13, which is 130.
2. Convert 11/10 by multiplying top and bottom by 13: 143/130.
3. Convert 13/13 by multiplying top and bottom by 10: 130/130.
4. Add the numerators: 143 + 130 = 273.
5. The resulting fraction is 273/130.
... |
Compute the quotient and remainder of 49971 divided by 52. Show the long division steps. | To divide 49971 by 52:
1. Determine how many times 52 goes into the leading digits of 49971.
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: 49971 / 52 = 960 with a remainder of 51.
Proof: (960 * 52) ... |
Compute the quotient and remainder of 88850 divided by 15. Show the long division steps. | To divide 88850 by 15:
1. Determine how many times 15 goes into the leading digits of 88850.
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: 88850 / 15 = 5923 with a remainder of 5.
Proof: (5923 * 15)... |
Multiply the following 2x2 matrices:
A =
[2, -2]
[3, 0]
B =
[0, 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) = (2*0) + (-2*1) = -2
C12 = (A11 * B12) + (A12 * B22) = (2*5) + (-2*2) = 6
C21 = (A21 * B11) + (A22 * B21) = (3*0) + (0*1) = 0
C22 = (A21 * B12) + (A22 * B22) = (3*5) + (0*2) = 15
Result Matrix... |
Add the fractions 4/10 and 1/6. 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 4/10 by multiplying top and bottom by 3: 12/30.
3. Convert 1/6 by multiplying top and bottom by 5: 5/30.
4. Add the numerators: 12 + 5 = 17.
5. The resulting fraction is 17/30.
Final Answer: 17/3... |
Add the fractions 13/3 and 6/13. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 3 and 13, which is 39.
2. Convert 13/3 by multiplying top and bottom by 13: 169/39.
3. Convert 6/13 by multiplying top and bottom by 3: 18/39.
4. Add the numerators: 169 + 18 = 187.
5. The resulting fraction is 187/39.
Final Answ... |
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 =
[3, -3]
[3, 5]
B =
[3, 4]
[0, -4] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (3*3) + (-3*0) = 9
C12 = (A11 * B12) + (A12 * B22) = (3*4) + (-3*-4) = 24
C21 = (A21 * B11) + (A22 * B21) = (3*3) + (5*0) = 9
C22 = (A21 * B12) + (A22 * B22) = (3*4) + (5*-4) = -8
Result Matr... |
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 |
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) |
Multiply the following 2x2 matrices:
A =
[5, -3]
[1, -5]
B =
[5, 0]
[-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) = (5*5) + (-3*-1) = 28
C12 = (A11 * B12) + (A12 * B22) = (5*0) + (-3*-3) = 9
C21 = (A21 * B11) + (A22 * B21) = (1*5) + (-5*-1) = 10
C22 = (A21 * B12) + (A22 * B22) = (1*0) + (-5*-3) = 15
Result... |
Add the fractions 8/5 and 8/8. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 5 and 8, which is 40.
2. Convert 8/5 by multiplying top and bottom by 8: 64/40.
3. Convert 8/8 by multiplying top and bottom by 5: 40/40.
4. Add the numerators: 64 + 40 = 104.
5. The resulting fraction is 104/40.
6. Simplify by d... |
In a right-angled triangle, the lengths of the two legs are a = 8 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: 8^2 + 3^2 = c^2
2. Calculate the squares: 64 + 9 = c^2
3. Sum the squares: 73 = c^2
4. Take the square root of both sides: c = √73
Final Answer: ... |
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 +... |
Compute the quotient and remainder of 50999 divided by 53. Show the long division steps. | To divide 50999 by 53:
1. Determine how many times 53 goes into the leading digits of 50999.
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: 50999 / 53 = 962 with a remainder of 13.
Proof: (962 * 53) ... |
Compute the indefinite integral of f(x) = 9x^1. | To find the indefinite integral ∫ 9x^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: 9 / 2 = 9/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 9x^1 dx = 9/2x^2 + 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 |
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 |
Add the fractions 13/11 and 12/7. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 11 and 7, which is 77.
2. Convert 13/11 by multiplying top and bottom by 7: 91/77.
3. Convert 12/7 by multiplying top and bottom by 11: 132/77.
4. Add the numerators: 91 + 132 = 223.
5. The resulting fraction is 223/77.
Final Ans... |
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... |
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 ... |
In a right-angled triangle, the lengths of the two legs are a = 13 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: 13^2 + 5^2 = c^2
2. Calculate the squares: 169 + 25 = c^2
3. Sum the squares: 194 = c^2
4. Take the square root of both sides: c = √194
Final Ans... |
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 13/6 and 13/11. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 6 and 11, which is 66.
2. Convert 13/6 by multiplying top and bottom by 11: 143/66.
3. Convert 13/11 by multiplying top and bottom by 6: 78/66.
4. Add the numerators: 143 + 78 = 221.
5. The resulting fraction is 221/66.
Final Ans... |
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 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 quotient and remainder of 32436 divided by 72. Show the long division steps. | To divide 32436 by 72:
1. Determine how many times 72 goes into the leading digits of 32436.
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: 32436 / 72 = 450 with a remainder of 36.
Proof: (450 * 72) ... |
Find the roots of the quadratic equation: 10x^2 + 1x + -7 = 0 | 1. Identify coefficients: a=10, b=1, c=-7.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1 - -280 = 281.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-1 ± √281) / 20.
5. Calculate roots: x1 = 0.7881527307120105, x2 = -0.8881527307120105.
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) = 8x^3. | To find the indefinite integral ∫ 8x^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: 8 / 4 = 2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 8x^3 dx = 2x^4 + C |
Compute the indefinite integral of f(x) = 1x^2. | To find the indefinite integral ∫ 1x^2 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 2 + 1 = 3.
2. Divide the coefficient by the new exponent: 1 / 3 = 1/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 1x^2 dx = 1/3x^3 + C |
Evaluate sin(0°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(0°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.0000 (exact form depends on root simplification if applicable) |
Compute the quotient and remainder of 64106 divided by 35. Show the long division steps. | To divide 64106 by 35:
1. Determine how many times 35 goes into the leading digits of 64106.
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: 64106 / 35 = 1831 with a remainder of 21.
Proof: (1831 * 35... |
Add the fractions 7/7 and 15/9. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 7 and 9, which is 63.
2. Convert 7/7 by multiplying top and bottom by 9: 63/63.
3. Convert 15/9 by multiplying top and bottom by 7: 105/63.
4. Add the numerators: 63 + 105 = 168.
5. The resulting fraction is 168/63.
6. Simplify b... |
Calculate the mean, population variance, and standard deviation for the dataset: [5, 1, 2, 14, 20] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (5 + 1 + 2 + 14 + 20) / 5 = 42 / 5 = 8.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-3.4, -7.4, -6.4, 5.6, 11.6]
Squared Differences: [11.56, 54.76, 40.96, 31.36, 134.56... |
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 |
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) = 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 |
Multiply the following 2x2 matrices:
A =
[2, 0]
[-2, -3]
B =
[-3, -3]
[-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*-3) + (0*-5) = -6
C12 = (A11 * B12) + (A12 * B22) = (2*-3) + (0*3) = -6
C21 = (A21 * B11) + (A22 * B21) = (-2*-3) + (-3*-5) = 21
C22 = (A21 * B12) + (A22 * B22) = (-2*-3) + (-3*3) = -3
Res... |
Evaluate the logarithm: log_4(256) | To evaluate log_4(256), we ask the question: '4 raised to what power equals 256?'
Let x be the unknown power: 4^x = 256
Since 4^4 = 256, it follows that x = 4.
Final Answer: 4 |
Calculate the mean, population variance, and standard deviation for the dataset: [18, 10, 19, 17, 12] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (18 + 10 + 19 + 17 + 12) / 5 = 76 / 5 = 15.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [2.8, -5.2, 3.8, 1.8, -3.2]
Squared Differences: [7.84, 27.04, 14.44, 3.24, 10.24]... |
Multiply the following 2x2 matrices:
A =
[-3, 4]
[-2, -4]
B =
[3, -5]
[-5, 5] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-3*3) + (4*-5) = -29
C12 = (A11 * B12) + (A12 * B22) = (-3*-5) + (4*5) = 35
C21 = (A21 * B11) + (A22 * B21) = (-2*3) + (-4*-5) = 14
C22 = (A21 * B12) + (A22 * B22) = (-2*-5) + (-4*5) = -10
R... |
Calculate the mean, population variance, and standard deviation for the dataset: [7, 6, 10, 15, 13] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (7 + 6 + 10 + 15 + 13) / 5 = 51 / 5 = 10.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-3.2, -4.2, -0.2, 4.8, 2.8]
Squared Differences: [10.24, 17.64, 0.04, 23.04, 7.84]
... |
Multiply the following 2x2 matrices:
A =
[0, 4]
[-5, -2]
B =
[2, -1]
[-5, 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) = (0*2) + (4*-5) = -20
C12 = (A11 * B12) + (A12 * B22) = (0*-1) + (4*4) = 16
C21 = (A21 * B11) + (A22 * B21) = (-5*2) + (-2*-5) = 0
C22 = (A21 * B12) + (A22 * B22) = (-5*-1) + (-2*4) = -3
Resul... |
Compute the indefinite integral of f(x) = 6x^1. | To find the indefinite integral ∫ 6x^1 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 1 + 1 = 2.
2. Divide the coefficient by the new exponent: 6 / 2 = 3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 6x^1 dx = 3x^2 + C |
Compute the quotient and remainder of 59892 divided by 42. Show the long division steps. | To divide 59892 by 42:
1. Determine how many times 42 goes into the leading digits of 59892.
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: 59892 / 42 = 1426 with a remainder of 0.
Proof: (1426 * 42)... |
Find the roots of the quadratic equation: 2x^2 + -9x + 9 = 0 | 1. Identify coefficients: a=2, b=-9, c=9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 81 - 72 = 9.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (9 ± √9) / 4.
5. Calculate roots: x1 = 3.0, x2 = 1.5.
Final Answer: x = 3.0, 1.5 |
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 +... |
Find the roots of the quadratic equation: 4x^2 + 1x + 7 = 0 | 1. Identify coefficients: a=4, b=1, c=7.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1^2 - 4(4)(7) = 1 - 112 = -111.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-1 ± √111i) / 8.
Final Answer: x = -0.125 ± 1.3169567191065923i |
Compute the quotient and remainder of 54271 divided by 21. Show the long division steps. | To divide 54271 by 21:
1. Determine how many times 21 goes into the leading digits of 54271.
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: 54271 / 21 = 2584 with a remainder of 7.
Proof: (2584 * 21)... |
Multiply the following 2x2 matrices:
A =
[-4, 2]
[-2, 0]
B =
[1, 0]
[-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) = (-4*1) + (2*-3) = -10
C12 = (A11 * B12) + (A12 * B22) = (-4*0) + (2*4) = 8
C21 = (A21 * B11) + (A22 * B21) = (-2*1) + (0*-3) = -2
C22 = (A21 * B12) + (A22 * B22) = (-2*0) + (0*4) = 0
Result M... |
Multiply the following 2x2 matrices:
A =
[-1, -1]
[-3, -2]
B =
[0, -1]
[-3, 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) = (-1*0) + (-1*-3) = 3
C12 = (A11 * B12) + (A12 * B22) = (-1*-1) + (-1*2) = -1
C21 = (A21 * B11) + (A22 * B21) = (-3*0) + (-2*-3) = 6
C22 = (A21 * B12) + (A22 * B22) = (-3*-1) + (-2*2) = -1
Res... |
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 +... |
Compute the quotient and remainder of 70542 divided by 38. Show the long division steps. | To divide 70542 by 38:
1. Determine how many times 38 goes into the leading digits of 70542.
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: 70542 / 38 = 1856 with a remainder of 14.
Proof: (1856 * 38... |
Calculate the mean, population variance, and standard deviation for the dataset: [5, 13, 5, 5, 6] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (5 + 13 + 5 + 5 + 6) / 5 = 34 / 5 = 6.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-1.8, 6.2, -1.8, -1.8, -0.8]
Squared Differences: [3.24, 38.44, 3.24, 3.24, 0.64]
S... |
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 |
Compute the indefinite integral of f(x) = 9x^1. | To find the indefinite integral ∫ 9x^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: 9 / 2 = 9/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 9x^1 dx = 9/2x^2 + C |
Add the fractions 12/13 and 9/15. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 13 and 15, which is 195.
2. Convert 12/13 by multiplying top and bottom by 15: 180/195.
3. Convert 9/15 by multiplying top and bottom by 13: 117/195.
4. Add the numerators: 180 + 117 = 297.
5. The resulting fraction is 297/195.
6... |
Compute the quotient and remainder of 23247 divided by 90. Show the long division steps. | To divide 23247 by 90:
1. Determine how many times 90 goes into the leading digits of 23247.
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: 23247 / 90 = 258 with a remainder of 27.
Proof: (258 * 90) ... |
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) = 5x^4. | To find the indefinite integral ∫ 5x^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: 5 / 5 = 1.
3. Combine and add the constant of integration C.
Final Answer: ∫ 5x^4 dx = 1x^5 + C |
Multiply the following 2x2 matrices:
A =
[5, 3]
[4, 2]
B =
[2, -1]
[5, -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*2) + (3*5) = 25
C12 = (A11 * B12) + (A12 * B22) = (5*-1) + (3*-4) = -17
C21 = (A21 * B11) + (A22 * B21) = (4*2) + (2*5) = 18
C22 = (A21 * B12) + (A22 * B22) = (4*-1) + (2*-4) = -12
Result ... |
Calculate the mean, population variance, and standard deviation for the dataset: [11, 10, 1, 10, 8] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (11 + 10 + 1 + 10 + 8) / 5 = 40 / 5 = 8.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [3.0, 2.0, -7.0, 2.0, 0.0]
Squared Differences: [9.0, 4.0, 49.0, 4.0, 0.0]
Sum of ... |
Compute the quotient and remainder of 1418 divided by 55. Show the long division steps. | To divide 1418 by 55:
1. Determine how many times 55 goes into the leading digits of 1418.
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: 1418 / 55 = 25 with a remainder of 43.
Proof: (25 * 55) + 43 ... |
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 |
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