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In a right-angled triangle, the lengths of the two legs are a = 6 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: 6^2 + 16^2 = c^2
2. Calculate the squares: 36 + 256 = c^2
3. Sum the squares: 292 = c^2
4. Take the square root of both sides: c = √292
Final Ans... |
In a right-angled triangle, the lengths of the two legs are a = 7 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: 7^2 + 3^2 = c^2
2. Calculate the squares: 49 + 9 = c^2
3. Sum the squares: 58 = c^2
4. Take the square root of both sides: c = √58
Final Answer: ... |
Calculate the mean, population variance, and standard deviation for the dataset: [6, 19, 16, 11, 2] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (6 + 19 + 16 + 11 + 2) / 5 = 54 / 5 = 10.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-4.8, 8.2, 5.2, 0.2, -8.8]
Squared Differences: [23.04, 67.24, 27.04, 0.04, 77.44]
... |
Compute the indefinite integral of f(x) = 2x^4. | To find the indefinite integral ∫ 2x^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: 2 / 5 = 2/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 2x^4 dx = 2/5x^5 + C |
Add the fractions 1/8 and 6/15. 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 1/8 by multiplying top and bottom by 15: 15/120.
3. Convert 6/15 by multiplying top and bottom by 8: 48/120.
4. Add the numerators: 15 + 48 = 63.
5. The resulting fraction is 63/120.
6. Simplify... |
Calculate the mean, population variance, and standard deviation for the dataset: [17, 5, 17, 4, 6] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (17 + 5 + 17 + 4 + 6) / 5 = 49 / 5 = 9.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [7.2, -4.8, 7.2, -5.8, -3.8]
Squared Differences: [51.84, 23.04, 51.84, 33.64, 14.44]
... |
Find the roots of the quadratic equation: 3x^2 + 7x + -5 = 0 | 1. Identify coefficients: a=3, b=7, c=-5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 49 - -60 = 109.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-7 ± √109) / 6.
5. Calculate roots: x1 = 0.5733844181517584, x2 = -2.906717751485092.
Final An... |
Add the fractions 1/13 and 3/7. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 13 and 7, which is 91.
2. Convert 1/13 by multiplying top and bottom by 7: 7/91.
3. Convert 3/7 by multiplying top and bottom by 13: 39/91.
4. Add the numerators: 7 + 39 = 46.
5. The resulting fraction is 46/91.
Final Answer: 46/... |
Add the fractions 15/14 and 13/14. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 14 and 14, which is 14.
2. Convert 15/14 by multiplying top and bottom by 1: 15/14.
3. Convert 13/14 by multiplying top and bottom by 1: 13/14.
4. Add the numerators: 15 + 13 = 28.
5. The resulting fraction is 28/14.
6. Simplify ... |
Compute the quotient and remainder of 9871 divided by 84. Show the long division steps. | To divide 9871 by 84:
1. Determine how many times 84 goes into the leading digits of 9871.
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: 9871 / 84 = 117 with a remainder of 43.
Proof: (117 * 84) + 4... |
In a right-angled triangle, the lengths of the two legs are a = 5 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: 5^2 + 12^2 = c^2
2. Calculate the squares: 25 + 144 = c^2
3. Sum the squares: 169 = c^2
4. Take the square root of both sides: c = √169
Final Ans... |
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) |
Multiply the following 2x2 matrices:
A =
[-2, 2]
[-5, -4]
B =
[-5, 4]
[0, 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*-5) + (2*0) = 10
C12 = (A11 * B12) + (A12 * B22) = (-2*4) + (2*1) = -6
C21 = (A21 * B11) + (A22 * B21) = (-5*-5) + (-4*0) = 25
C22 = (A21 * B12) + (A22 * B22) = (-5*4) + (-4*1) = -24
Resu... |
Calculate the mean, population variance, and standard deviation for the dataset: [7, 19, 17, 3, 6] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (7 + 19 + 17 + 3 + 6) / 5 = 52 / 5 = 10.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-3.4, 8.6, 6.6, -7.4, -4.4]
Squared Differences: [11.56, 73.96, 43.56, 54.76, 19.36]... |
Find the roots of the quadratic equation: 7x^2 + -10x + -10 = 0 | 1. Identify coefficients: a=7, b=-10, c=-10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 100 - -280 = 380.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (10 ± √380) / 14.
5. Calculate roots: x1 = 2.1066849064012803, x2 = -0.6781134778298519.
F... |
Compute the quotient and remainder of 9118 divided by 29. Show the long division steps. | To divide 9118 by 29:
1. Determine how many times 29 goes into the leading digits of 9118.
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: 9118 / 29 = 314 with a remainder of 12.
Proof: (314 * 29) + 1... |
Add the fractions 15/10 and 8/12. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 10 and 12, which is 60.
2. Convert 15/10 by multiplying top and bottom by 6: 90/60.
3. Convert 8/12 by multiplying top and bottom by 5: 40/60.
4. Add the numerators: 90 + 40 = 130.
5. The resulting fraction is 130/60.
6. Simplify... |
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^4. | To find the indefinite integral ∫ 10x^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: 10 / 5 = 2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 10x^4 dx = 2x^5 + C |
Evaluate sin(90°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(90°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 1.0000 (exact form depends on root simplification if applicable) |
In a right-angled triangle, the lengths of the two legs are a = 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... |
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 |
Find the roots of the quadratic equation: 10x^2 + 0x + -5 = 0 | 1. Identify coefficients: a=10, b=0, c=-5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 0 - -200 = 200.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (0 ± √200) / 20.
5. Calculate roots: x1 = 0.7071067811865476, x2 = -0.7071067811865476.
Final ... |
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 indefinite integral of f(x) = 3x^2. | To find the indefinite integral ∫ 3x^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: 3 / 3 = 1.
3. Combine and add the constant of integration C.
Final Answer: ∫ 3x^2 dx = 1x^3 + C |
In a right-angled triangle, the lengths of the two legs are a = 11 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: 11^2 + 12^2 = c^2
2. Calculate the squares: 121 + 144 = c^2
3. Sum the squares: 265 = c^2
4. Take the square root of both sides: c = √265
Final A... |
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 the logarithm: log_3(81) | To evaluate log_3(81), we ask the question: '3 raised to what power equals 81?'
Let x be the unknown power: 3^x = 81
Since 3^4 = 81, it follows that x = 4.
Final Answer: 4 |
Compute the indefinite integral of f(x) = 2x^5. | To find the indefinite integral ∫ 2x^5 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 5 + 1 = 6.
2. Divide the coefficient by the new exponent: 2 / 6 = 1/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 2x^5 dx = 1/3x^6 + C |
Multiply the following 2x2 matrices:
A =
[1, -5]
[-5, -1]
B =
[-1, -1]
[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*-1) + (-5*2) = -11
C12 = (A11 * B12) + (A12 * B22) = (1*-1) + (-5*5) = -26
C21 = (A21 * B11) + (A22 * B21) = (-5*-1) + (-1*2) = 3
C22 = (A21 * B12) + (A22 * B22) = (-5*-1) + (-1*5) = 0
Res... |
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 |
Multiply the following 2x2 matrices:
A =
[4, -2]
[4, -1]
B =
[-3, -4]
[-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) = (4*-3) + (-2*-1) = -10
C12 = (A11 * B12) + (A12 * B22) = (4*-4) + (-2*-5) = -6
C21 = (A21 * B11) + (A22 * B21) = (4*-3) + (-1*-1) = -11
C22 = (A21 * B12) + (A22 * B22) = (4*-4) + (-1*-5) = -11... |
Add the fractions 13/13 and 6/4. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 13 and 4, which is 52.
2. Convert 13/13 by multiplying top and bottom by 4: 52/52.
3. Convert 6/4 by multiplying top and bottom by 13: 78/52.
4. Add the numerators: 52 + 78 = 130.
5. The resulting fraction is 130/52.
6. Simplify ... |
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 indefinite integral of f(x) = 3x^5. | To find the indefinite integral ∫ 3x^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: 3 / 6 = 1/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 3x^5 dx = 1/2x^6 + C |
Compute the quotient and remainder of 14948 divided by 91. Show the long division steps. | To divide 14948 by 91:
1. Determine how many times 91 goes into the leading digits of 14948.
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: 14948 / 91 = 164 with a remainder of 24.
Proof: (164 * 91) ... |
Calculate the mean, population variance, and standard deviation for the dataset: [3, 13, 11, 15, 18] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (3 + 13 + 11 + 15 + 18) / 5 = 60 / 5 = 12.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-9.0, 1.0, -1.0, 3.0, 6.0]
Squared Differences: [81.0, 1.0, 1.0, 9.0, 36.0]
Sum... |
In a right-angled triangle, the lengths of the two legs are a = 15 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: 15^2 + 8^2 = c^2
2. Calculate the squares: 225 + 64 = c^2
3. Sum the squares: 289 = c^2
4. Take the square root of both sides: c = √289
Final Ans... |
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 + 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 ... |
Add the fractions 4/13 and 7/4. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 13 and 4, which is 52.
2. Convert 4/13 by multiplying top and bottom by 4: 16/52.
3. Convert 7/4 by multiplying top and bottom by 13: 91/52.
4. Add the numerators: 16 + 91 = 107.
5. The resulting fraction is 107/52.
Final Answer:... |
Multiply the following 2x2 matrices:
A =
[-2, -5]
[1, 3]
B =
[-2, -1]
[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*-2) + (-5*4) = -16
C12 = (A11 * B12) + (A12 * B22) = (-2*-1) + (-5*-3) = 17
C21 = (A21 * B11) + (A22 * B21) = (1*-2) + (3*4) = 10
C22 = (A21 * B12) + (A22 * B22) = (1*-1) + (3*-3) = -10
R... |
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) |
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 |
Find the roots of the quadratic equation: 10x^2 + -4x + -4 = 0 | 1. Identify coefficients: a=10, b=-4, c=-4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 16 - -160 = 176.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (4 ± √176) / 20.
5. Calculate roots: x1 = 0.86332495807108, x2 = -0.46332495807108.
Final An... |
Add the fractions 13/7 and 4/5. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 7 and 5, which is 35.
2. Convert 13/7 by multiplying top and bottom by 5: 65/35.
3. Convert 4/5 by multiplying top and bottom by 7: 28/35.
4. Add the numerators: 65 + 28 = 93.
5. The resulting fraction is 93/35.
Final Answer: 93/... |
Multiply the following 2x2 matrices:
A =
[-4, 0]
[-2, -4]
B =
[-5, -3]
[-4, -4] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-4*-5) + (0*-4) = 20
C12 = (A11 * B12) + (A12 * B22) = (-4*-3) + (0*-4) = 12
C21 = (A21 * B11) + (A22 * B21) = (-2*-5) + (-4*-4) = 26
C22 = (A21 * B12) + (A22 * B22) = (-2*-3) + (-4*-4) = 22
... |
Add the fractions 5/4 and 9/14. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 4 and 14, which is 28.
2. Convert 5/4 by multiplying top and bottom by 7: 35/28.
3. Convert 9/14 by multiplying top and bottom by 2: 18/28.
4. Add the numerators: 35 + 18 = 53.
5. The resulting fraction is 53/28.
Final Answer: 53... |
Compute the quotient and remainder of 75048 divided by 34. Show the long division steps. | To divide 75048 by 34:
1. Determine how many times 34 goes into the leading digits of 75048.
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: 75048 / 34 = 2207 with a remainder of 10.
Proof: (2207 * 34... |
Calculate the mean, population variance, and standard deviation for the dataset: [9, 18, 14, 19, 15] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (9 + 18 + 14 + 19 + 15) / 5 = 75 / 5 = 15.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-6.0, 3.0, -1.0, 4.0, 0.0]
Squared Differences: [36.0, 9.0, 1.0, 16.0, 0.0]
Sum... |
Add the fractions 4/14 and 7/6. 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 4/14 by multiplying top and bottom by 3: 12/42.
3. Convert 7/6 by multiplying top and bottom by 7: 49/42.
4. Add the numerators: 12 + 49 = 61.
5. The resulting fraction is 61/42.
Final Answer: 61... |
In a right-angled triangle, the lengths of the two legs are a = 19 and b = 19. 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 + 19^2 = c^2
2. Calculate the squares: 361 + 361 = c^2
3. Sum the squares: 722 = c^2
4. Take the square root of both sides: c = √722
Final A... |
In a right-angled triangle, the lengths of the two legs are a = 9 and b = 7. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 9^2 + 7^2 = c^2
2. Calculate the squares: 81 + 49 = c^2
3. Sum the squares: 130 = c^2
4. Take the square root of both sides: c = √130
Final Answe... |
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 indefinite integral of f(x) = 9x^2. | To find the indefinite integral ∫ 9x^2 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 2 + 1 = 3.
2. Divide the coefficient by the new exponent: 9 / 3 = 3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 9x^2 dx = 3x^3 + C |
Add the fractions 1/8 and 6/5. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 8 and 5, which is 40.
2. Convert 1/8 by multiplying top and bottom by 5: 5/40.
3. Convert 6/5 by multiplying top and bottom by 8: 48/40.
4. Add the numerators: 5 + 48 = 53.
5. The resulting fraction is 53/40.
Final Answer: 53/40 |
Compute the quotient and remainder of 80574 divided by 85. Show the long division steps. | To divide 80574 by 85:
1. Determine how many times 85 goes into the leading digits of 80574.
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: 80574 / 85 = 947 with a remainder of 79.
Proof: (947 * 85) ... |
In a right-angled triangle, the lengths of the two legs are a = 14 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: 14^2 + 11^2 = c^2
2. Calculate the squares: 196 + 121 = c^2
3. Sum the squares: 317 = c^2
4. Take the square root of both sides: c = √317
Final A... |
Add the fractions 8/14 and 1/2. 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 8/14 by multiplying top and bottom by 1: 8/14.
3. Convert 1/2 by multiplying top and bottom by 7: 7/14.
4. Add the numerators: 8 + 7 = 15.
5. The resulting fraction is 15/14.
Final Answer: 15/14 |
Compute the quotient and remainder of 80219 divided by 68. Show the long division steps. | To divide 80219 by 68:
1. Determine how many times 68 goes into the leading digits of 80219.
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: 80219 / 68 = 1179 with a remainder of 47.
Proof: (1179 * 68... |
Compute the quotient and remainder of 45289 divided by 89. Show the long division steps. | To divide 45289 by 89:
1. Determine how many times 89 goes into the leading digits of 45289.
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: 45289 / 89 = 508 with a remainder of 77.
Proof: (508 * 89) ... |
Add the fractions 7/2 and 11/4. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 2 and 4, which is 4.
2. Convert 7/2 by multiplying top and bottom by 2: 14/4.
3. Convert 11/4 by multiplying top and bottom by 1: 11/4.
4. Add the numerators: 14 + 11 = 25.
5. The resulting fraction is 25/4.
Final Answer: 25/4 |
In a right-angled triangle, the lengths of the two legs are a = 14 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: 14^2 + 10^2 = c^2
2. Calculate the squares: 196 + 100 = c^2
3. Sum the squares: 296 = c^2
4. Take the square root of both sides: c = √296
Final A... |
Compute the indefinite integral of f(x) = 4x^1. | To find the indefinite integral ∫ 4x^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: 4 / 2 = 2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 4x^1 dx = 2x^2 + C |
Find the roots of the quadratic equation: 7x^2 + 5x + -7 = 0 | 1. Identify coefficients: a=7, b=5, c=-7.
2. Calculate the discriminant: Δ = b^2 - 4ac = 25 - -196 = 221.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-5 ± √221) / 14.
5. Calculate roots: x1 = 0.7047191962370362, x2 = -1.4190049105227505.
Final... |
Compute the indefinite integral of f(x) = 2x^5. | To find the indefinite integral ∫ 2x^5 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 5 + 1 = 6.
2. Divide the coefficient by the new exponent: 2 / 6 = 1/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 2x^5 dx = 1/3x^6 + C |
Evaluate the logarithm: log_3(9) | To evaluate log_3(9), we ask the question: '3 raised to what power equals 9?'
Let x be the unknown power: 3^x = 9
Since 3^2 = 9, it follows that x = 2.
Final Answer: 2 |
In a right-angled triangle, the lengths of the two legs are a = 15 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: 15^2 + 4^2 = c^2
2. Calculate the squares: 225 + 16 = c^2
3. Sum the squares: 241 = c^2
4. Take the square root of both sides: c = √241
Final Ans... |
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 |
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 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 |
Calculate the mean, population variance, and standard deviation for the dataset: [18, 15, 18, 3, 11] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (18 + 15 + 18 + 3 + 11) / 5 = 65 / 5 = 13.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [5.0, 2.0, 5.0, -10.0, -2.0]
Squared Differences: [25.0, 4.0, 25.0, 100.0, 4.0]
... |
Evaluate sin(45°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(45°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.7071 (exact form depends on root simplification if applicable) |
Add the fractions 13/6 and 4/7. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 6 and 7, which is 42.
2. Convert 13/6 by multiplying top and bottom by 7: 91/42.
3. Convert 4/7 by multiplying top and bottom by 6: 24/42.
4. Add the numerators: 91 + 24 = 115.
5. The resulting fraction is 115/42.
Final Answer: 1... |
Multiply the following 2x2 matrices:
A =
[-1, 4]
[-5, -4]
B =
[5, 3]
[-3, 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) = (-1*5) + (4*-3) = -17
C12 = (A11 * B12) + (A12 * B22) = (-1*3) + (4*0) = -3
C21 = (A21 * B11) + (A22 * B21) = (-5*5) + (-4*-3) = -13
C22 = (A21 * B12) + (A22 * B22) = (-5*3) + (-4*0) = -15
Re... |
Add the fractions 5/8 and 4/15. 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 5/8 by multiplying top and bottom by 15: 75/120.
3. Convert 4/15 by multiplying top and bottom by 8: 32/120.
4. Add the numerators: 75 + 32 = 107.
5. The resulting fraction is 107/120.
Final Ans... |
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) |
Calculate the mean, population variance, and standard deviation for the dataset: [16, 12, 8, 16, 8] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (16 + 12 + 8 + 16 + 8) / 5 = 60 / 5 = 12.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [4.0, 0.0, -4.0, 4.0, -4.0]
Squared Differences: [16.0, 0.0, 16.0, 16.0, 16.0]
Su... |
Evaluate the logarithm: log_3(9) | To evaluate log_3(9), we ask the question: '3 raised to what power equals 9?'
Let x be the unknown power: 3^x = 9
Since 3^2 = 9, it follows that x = 2.
Final Answer: 2 |
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 |
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 +... |
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 |
Find the roots of the quadratic equation: 9x^2 + -9x + 8 = 0 | 1. Identify coefficients: a=9, b=-9, c=8.
2. Calculate the discriminant: Δ = b^2 - 4ac = -9^2 - 4(9)(8) = 81 - 288 = -207.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (9 ± √207i) / 18.
Final Answer: x = 0.5 ± 0.7993052538854533i |
Compute the indefinite integral of f(x) = 7x^5. | To find the indefinite integral ∫ 7x^5 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 5 + 1 = 6.
2. Divide the coefficient by the new exponent: 7 / 6 = 7/6.
3. Combine and add the constant of integration C.
Final Answer: ∫ 7x^5 dx = 7/6x^6 + C |
Find the roots of the quadratic equation: 2x^2 + 5x + 8 = 0 | 1. Identify coefficients: a=2, b=5, c=8.
2. Calculate the discriminant: Δ = b^2 - 4ac = 5^2 - 4(2)(8) = 25 - 64 = -39.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-5 ± √39i) / 4.
Final Answer: x = -1.25 ± 1.5612494995995996i |
Multiply the following 2x2 matrices:
A =
[-1, -5]
[-4, 0]
B =
[4, -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) = (-1*4) + (-5*5) = -29
C12 = (A11 * B12) + (A12 * B22) = (-1*-1) + (-5*-2) = 11
C21 = (A21 * B11) + (A22 * B21) = (-4*4) + (0*5) = -16
C22 = (A21 * B12) + (A22 * B22) = (-4*-1) + (0*-2) = 4
Re... |
In a right-angled triangle, the lengths of the two legs are a = 12 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: 12^2 + 8^2 = c^2
2. Calculate the squares: 144 + 64 = c^2
3. Sum the squares: 208 = c^2
4. Take the square root of both sides: c = √208
Final Ans... |
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) |
Multiply the following 2x2 matrices:
A =
[4, -2]
[-5, 0]
B =
[-4, 0]
[-1, 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) = (4*-4) + (-2*-1) = -14
C12 = (A11 * B12) + (A12 * B22) = (4*0) + (-2*0) = 0
C21 = (A21 * B11) + (A22 * B21) = (-5*-4) + (0*-1) = 20
C22 = (A21 * B12) + (A22 * B22) = (-5*0) + (0*0) = 0
Result... |
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) |
Find the roots of the quadratic equation: 10x^2 + -7x + -3 = 0 | 1. Identify coefficients: a=10, b=-7, c=-3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 49 - -120 = 169.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (7 ± √169) / 20.
5. Calculate roots: x1 = 1.0, x2 = -0.3.
Final Answer: x = 1.0, -0.3 |
Calculate the mean, population variance, and standard deviation for the dataset: [9, 18, 11, 2, 3] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (9 + 18 + 11 + 2 + 3) / 5 = 43 / 5 = 8.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [0.4, 9.4, 2.4, -6.6, -5.6]
Squared Differences: [0.16, 88.36, 5.76, 43.56, 31.36]
... |
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 ... |
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 ... |
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... |
Find the roots of the quadratic equation: 7x^2 + -5x + 9 = 0 | 1. Identify coefficients: a=7, b=-5, c=9.
2. Calculate the discriminant: Δ = b^2 - 4ac = -5^2 - 4(7)(9) = 25 - 252 = -227.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (5 ± √227i) / 14.
Final Answer: x = 0.35714285714285715 ± 1.076179940951383i |
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 |
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... |
Add the fractions 12/3 and 12/14. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 3 and 14, which is 42.
2. Convert 12/3 by multiplying top and bottom by 14: 168/42.
3. Convert 12/14 by multiplying top and bottom by 3: 36/42.
4. Add the numerators: 168 + 36 = 204.
5. The resulting fraction is 204/42.
6. Simpli... |
Calculate the mean, population variance, and standard deviation for the dataset: [18, 18, 14, 18, 2] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (18 + 18 + 14 + 18 + 2) / 5 = 70 / 5 = 14.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [4.0, 4.0, 0.0, 4.0, -12.0]
Squared Differences: [16.0, 16.0, 0.0, 16.0, 144.0]
... |
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