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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... |
Multiply the following 2x2 matrices:
A =
[-4, -5]
[-1, -4]
B =
[0, -3]
[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) + (-5*4) = -20
C12 = (A11 * B12) + (A12 * B22) = (-4*-3) + (-5*-3) = 27
C21 = (A21 * B11) + (A22 * B21) = (-1*0) + (-4*4) = -16
C22 = (A21 * B12) + (A22 * B22) = (-1*-3) + (-4*-3) = 15
... |
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 |
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... |
Multiply the following 2x2 matrices:
A =
[-1, -4]
[0, -2]
B =
[-5, -5]
[1, 1] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-1*-5) + (-4*1) = 1
C12 = (A11 * B12) + (A12 * B22) = (-1*-5) + (-4*1) = 1
C21 = (A21 * B11) + (A22 * B21) = (0*-5) + (-2*1) = -2
C22 = (A21 * B12) + (A22 * B22) = (0*-5) + (-2*1) = -2
Resul... |
Add the fractions 8/9 and 3/15. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 9 and 15, which is 45.
2. Convert 8/9 by multiplying top and bottom by 5: 40/45.
3. Convert 3/15 by multiplying top and bottom by 3: 9/45.
4. Add the numerators: 40 + 9 = 49.
5. The resulting fraction is 49/45.
Final Answer: 49/4... |
Add the fractions 7/8 and 1/14. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 8 and 14, which is 56.
2. Convert 7/8 by multiplying top and bottom by 7: 49/56.
3. Convert 1/14 by multiplying top and bottom by 4: 4/56.
4. Add the numerators: 49 + 4 = 53.
5. The resulting fraction is 53/56.
Final Answer: 53/5... |
Find the derivative of f(x) = (2x + 3)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 2x + 3.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 2x + 3. Its derivative is h'(x) = 2.
4. Multiply g'(u) by h'(x): f'(x) = 3(2x + 3)^2 * 2 = 6(2x +... |
Multiply the following 2x2 matrices:
A =
[0, 1]
[4, 2]
B =
[-5, 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) = (0*-5) + (1*-1) = -1
C12 = (A11 * B12) + (A12 * B22) = (0*5) + (1*2) = 2
C21 = (A21 * B11) + (A22 * B21) = (4*-5) + (2*-1) = -22
C22 = (A21 * B12) + (A22 * B22) = (4*5) + (2*2) = 24
Result Ma... |
In a right-angled triangle, the lengths of the two legs are a = 8 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: 8^2 + 13^2 = c^2
2. Calculate the squares: 64 + 169 = c^2
3. Sum the squares: 233 = c^2
4. Take the square root of both sides: c = √233
Final Ans... |
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 47402 divided by 69. Show the long division steps. | To divide 47402 by 69:
1. Determine how many times 69 goes into the leading digits of 47402.
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: 47402 / 69 = 686 with a remainder of 68.
Proof: (686 * 69) ... |
In a right-angled triangle, the lengths of the two legs are a = 5 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: 5^2 + 13^2 = c^2
2. Calculate the squares: 25 + 169 = c^2
3. Sum the squares: 194 = c^2
4. Take the square root of both sides: c = √194
Final Ans... |
Find the roots of the quadratic equation: 3x^2 + -5x + 7 = 0 | 1. Identify coefficients: a=3, b=-5, c=7.
2. Calculate the discriminant: Δ = b^2 - 4ac = -5^2 - 4(3)(7) = 25 - 84 = -59.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (5 ± √59i) / 6.
Final Answer: x = 0.8333333333333334 ± 1.2801909579781012i |
In a right-angled triangle, the lengths of the two legs are a = 8 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: 8^2 + 17^2 = c^2
2. Calculate the squares: 64 + 289 = c^2
3. Sum the squares: 353 = c^2
4. Take the square root of both sides: c = √353
Final Ans... |
In a right-angled triangle, the lengths of the two legs are a = 12 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: 12^2 + 20^2 = c^2
2. Calculate the squares: 144 + 400 = c^2
3. Sum the squares: 544 = c^2
4. Take the square root of both sides: c = √544
Final A... |
Multiply the following 2x2 matrices:
A =
[3, 3]
[4, 1]
B =
[1, 3]
[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) = (3*1) + (3*3) = 12
C12 = (A11 * B12) + (A12 * B22) = (3*3) + (3*-2) = 3
C21 = (A21 * B11) + (A22 * B21) = (4*1) + (1*3) = 7
C22 = (A21 * B12) + (A22 * B22) = (4*3) + (1*-2) = 10
Result Matrix... |
Find the roots of the quadratic equation: 2x^2 + 3x + 9 = 0 | 1. Identify coefficients: a=2, b=3, c=9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 3^2 - 4(2)(9) = 9 - 72 = -63.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-3 ± √63i) / 4.
Final Answer: x = -0.75 ± 1.984313483298443i |
Find the roots of the quadratic equation: 6x^2 + -3x + 2 = 0 | 1. Identify coefficients: a=6, b=-3, c=2.
2. Calculate the discriminant: Δ = b^2 - 4ac = -3^2 - 4(6)(2) = 9 - 48 = -39.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (3 ± √39i) / 12.
Final Answer: x = 0.25 ± 0.5204164998665332i |
Compute the indefinite integral of f(x) = 10x^5. | To find the indefinite integral ∫ 10x^5 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 5 + 1 = 6.
2. Divide the coefficient by the new exponent: 10 / 6 = 5/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 10x^5 dx = 5/3x^6 + C |
Add the fractions 3/12 and 11/11. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 12 and 11, which is 132.
2. Convert 3/12 by multiplying top and bottom by 11: 33/132.
3. Convert 11/11 by multiplying top and bottom by 12: 132/132.
4. Add the numerators: 33 + 132 = 165.
5. The resulting fraction is 165/132.
6. ... |
Find the derivative of f(x) = (4x + 3)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 4x + 3.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 3. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 3)^2 * 4 = 12(4x ... |
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 +... |
Add the fractions 4/7 and 6/2. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 7 and 2, which is 14.
2. Convert 4/7 by multiplying top and bottom by 2: 8/14.
3. Convert 6/2 by multiplying top and bottom by 7: 42/14.
4. Add the numerators: 8 + 42 = 50.
5. The resulting fraction is 50/14.
6. Simplify by divid... |
Multiply the following 2x2 matrices:
A =
[-2, -2]
[2, -4]
B =
[-2, -3]
[2, 0] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-2*-2) + (-2*2) = 0
C12 = (A11 * B12) + (A12 * B22) = (-2*-3) + (-2*0) = 6
C21 = (A21 * B11) + (A22 * B21) = (2*-2) + (-4*2) = -12
C22 = (A21 * B12) + (A22 * B22) = (2*-3) + (-4*0) = -6
Resu... |
Calculate the mean, population variance, and standard deviation for the dataset: [6, 7, 12, 20, 6] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (6 + 7 + 12 + 20 + 6) / 5 = 51 / 5 = 10.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-4.2, -3.2, 1.8, 9.8, -4.2]
Squared Differences: [17.64, 10.24, 3.24, 96.04, 17.64]
... |
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) |
Calculate the mean, population variance, and standard deviation for the dataset: [15, 4, 19, 20, 13] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (15 + 4 + 19 + 20 + 13) / 5 = 71 / 5 = 14.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [0.8, -10.2, 4.8, 5.8, -1.2]
Squared Differences: [0.64, 104.04, 23.04, 33.64, 1.44... |
Find the derivative of f(x) = (5x + 2)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 5x + 2.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 5x + 2. Its derivative is h'(x) = 5.
4. Multiply g'(u) by h'(x): f'(x) = 3(5x + 2)^2 * 5 = 15(5x ... |
Compute the quotient and remainder of 66674 divided by 81. Show the long division steps. | To divide 66674 by 81:
1. Determine how many times 81 goes into the leading digits of 66674.
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: 66674 / 81 = 823 with a remainder of 11.
Proof: (823 * 81) ... |
Calculate the mean, population variance, and standard deviation for the dataset: [1, 12, 20, 5, 3] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (1 + 12 + 20 + 5 + 3) / 5 = 41 / 5 = 8.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-7.2, 3.8, 11.8, -3.2, -5.2]
Squared Differences: [51.84, 14.44, 139.24, 10.24, 27.04... |
Find the roots of the quadratic equation: 10x^2 + 2x + 5 = 0 | 1. Identify coefficients: a=10, b=2, c=5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 2^2 - 4(10)(5) = 4 - 200 = -196.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-2 ± √196i) / 20.
Final Answer: x = -0.1 ± 0.7i |
Calculate the mean, population variance, and standard deviation for the dataset: [1, 17, 14, 15, 1] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (1 + 17 + 14 + 15 + 1) / 5 = 48 / 5 = 9.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-8.6, 7.4, 4.4, 5.4, -8.6]
Squared Differences: [73.96, 54.76, 19.36, 29.16, 73.96]
... |
Multiply the following 2x2 matrices:
A =
[0, 5]
[-5, 5]
B =
[0, -1]
[-2, -3] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (0*0) + (5*-2) = -10
C12 = (A11 * B12) + (A12 * B22) = (0*-1) + (5*-3) = -15
C21 = (A21 * B11) + (A22 * B21) = (-5*0) + (5*-2) = -10
C22 = (A21 * B12) + (A22 * B22) = (-5*-1) + (5*-3) = -10
R... |
Multiply the following 2x2 matrices:
A =
[-1, 4]
[-2, -1]
B =
[5, -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) = (-1*5) + (4*4) = 11
C12 = (A11 * B12) + (A12 * B22) = (-1*-1) + (4*3) = 13
C21 = (A21 * B11) + (A22 * B21) = (-2*5) + (-1*4) = -14
C22 = (A21 * B12) + (A22 * B22) = (-2*-1) + (-1*3) = -1
Resu... |
Compute the quotient and remainder of 18242 divided by 63. Show the long division steps. | To divide 18242 by 63:
1. Determine how many times 63 goes into the leading digits of 18242.
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: 18242 / 63 = 289 with a remainder of 35.
Proof: (289 * 63) ... |
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 |
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 |
Compute the quotient and remainder of 11925 divided by 29. Show the long division steps. | To divide 11925 by 29:
1. Determine how many times 29 goes into the leading digits of 11925.
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: 11925 / 29 = 411 with a remainder of 6.
Proof: (411 * 29) +... |
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 +... |
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 +... |
Evaluate sin(60°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(60°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.8660 (exact form depends on root simplification if applicable) |
Evaluate cos(45°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(45°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.7071 (exact form depends on root simplification if applicable) |
Find the roots of the quadratic equation: 7x^2 + 1x + -7 = 0 | 1. Identify coefficients: a=7, b=1, c=-7.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1 - -196 = 197.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-1 ± √197) / 14.
5. Calculate roots: x1 = 0.9311192034012999, x2 = -1.0739763462584428.
Final ... |
Compute the quotient and remainder of 75215 divided by 85. Show the long division steps. | To divide 75215 by 85:
1. Determine how many times 85 goes into the leading digits of 75215.
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: 75215 / 85 = 884 with a remainder of 75.
Proof: (884 * 85) ... |
Calculate the mean, population variance, and standard deviation for the dataset: [4, 17, 8, 5, 4] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (4 + 17 + 8 + 5 + 4) / 5 = 38 / 5 = 7.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-3.6, 9.4, 0.4, -2.6, -3.6]
Squared Differences: [12.96, 88.36, 0.16, 6.76, 12.96]
... |
Calculate the mean, population variance, and standard deviation for the dataset: [2, 2, 14, 2, 6] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (2 + 2 + 14 + 2 + 6) / 5 = 26 / 5 = 5.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-3.2, -3.2, 8.8, -3.2, 0.8]
Squared Differences: [10.24, 10.24, 77.44, 10.24, 0.64]
... |
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 +... |
Calculate the mean, population variance, and standard deviation for the dataset: [13, 11, 18, 16, 5] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (13 + 11 + 18 + 16 + 5) / 5 = 63 / 5 = 12.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [0.4, -1.6, 5.4, 3.4, -7.6]
Squared Differences: [0.16, 2.56, 29.16, 11.56, 57.76]
... |
Find the derivative of f(x) = (3x + 5)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 3x + 5.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 5. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 5)^2 * 3 = 9(3x +... |
Calculate the mean, population variance, and standard deviation for the dataset: [19, 6, 4, 9, 1] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (19 + 6 + 4 + 9 + 1) / 5 = 39 / 5 = 7.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [11.2, -1.8, -3.8, 1.2, -6.8]
Squared Differences: [125.44, 3.24, 14.44, 1.44, 46.24]
... |
Multiply the following 2x2 matrices:
A =
[-4, -4]
[3, 2]
B =
[5, -1]
[-2, 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) + (-4*-2) = -12
C12 = (A11 * B12) + (A12 * B22) = (-4*-1) + (-4*4) = -12
C21 = (A21 * B11) + (A22 * B21) = (3*5) + (2*-2) = 11
C22 = (A21 * B12) + (A22 * B22) = (3*-1) + (2*4) = 5
Resu... |
Find the derivative of f(x) = (4x + 2)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 4x + 2.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 2. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 2)^2 * 4 = 12(4x ... |
In a right-angled triangle, the lengths of the two legs are a = 5 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: 5^2 + 11^2 = c^2
2. Calculate the squares: 25 + 121 = c^2
3. Sum the squares: 146 = c^2
4. Take the square root of both sides: c = √146
Final Ans... |
Add the fractions 11/13 and 4/5. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 13 and 5, which is 65.
2. Convert 11/13 by multiplying top and bottom by 5: 55/65.
3. Convert 4/5 by multiplying top and bottom by 13: 52/65.
4. Add the numerators: 55 + 52 = 107.
5. The resulting fraction is 107/65.
Final Answer... |
Add the fractions 3/13 and 3/2. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 13 and 2, which is 26.
2. Convert 3/13 by multiplying top and bottom by 2: 6/26.
3. Convert 3/2 by multiplying top and bottom by 13: 39/26.
4. Add the numerators: 6 + 39 = 45.
5. The resulting fraction is 45/26.
Final Answer: 45/... |
Compute the quotient and remainder of 68679 divided by 88. Show the long division steps. | To divide 68679 by 88:
1. Determine how many times 88 goes into the leading digits of 68679.
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: 68679 / 88 = 780 with a remainder of 39.
Proof: (780 * 88) ... |
Multiply the following 2x2 matrices:
A =
[-1, 1]
[4, -4]
B =
[5, 0]
[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) = (-1*5) + (1*4) = -1
C12 = (A11 * B12) + (A12 * B22) = (-1*0) + (1*3) = 3
C21 = (A21 * B11) + (A22 * B21) = (4*5) + (-4*4) = 4
C22 = (A21 * B12) + (A22 * B22) = (4*0) + (-4*3) = -12
Result Mat... |
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 |
In a right-angled triangle, the lengths of the two legs are a = 20 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: 20^2 + 15^2 = c^2
2. Calculate the squares: 400 + 225 = c^2
3. Sum the squares: 625 = c^2
4. Take the square root of both sides: c = √625
Final A... |
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 = 4.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (10 ± √4) / 8.
5. Calculate roots: x1 = 1.5, x2 = 1.0.
Final Answer: x = 1.5, 1.0 |
Calculate the mean, population variance, and standard deviation for the dataset: [10, 10, 17, 15, 5] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (10 + 10 + 17 + 15 + 5) / 5 = 57 / 5 = 11.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-1.4, -1.4, 5.6, 3.6, -6.4]
Squared Differences: [1.96, 1.96, 31.36, 12.96, 40.96]... |
Evaluate cos(90°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(90°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.0000 (exact form depends on root simplification if applicable) |
Evaluate the logarithm: log_3(243) | To evaluate log_3(243), we ask the question: '3 raised to what power equals 243?'
Let x be the unknown power: 3^x = 243
Since 3^5 = 243, it follows that x = 5.
Final Answer: 5 |
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 |
Calculate the mean, population variance, and standard deviation for the dataset: [5, 7, 5, 11, 6] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (5 + 7 + 5 + 11 + 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, 0.2, -1.8, 4.2, -0.8]
Squared Differences: [3.24, 0.04, 3.24, 17.64, 0.64]
Su... |
Evaluate the logarithm: log_4(64) | To evaluate log_4(64), we ask the question: '4 raised to what power equals 64?'
Let x be the unknown power: 4^x = 64
Since 4^3 = 64, it follows that x = 3.
Final Answer: 3 |
Compute the quotient and remainder of 50229 divided by 79. Show the long division steps. | To divide 50229 by 79:
1. Determine how many times 79 goes into the leading digits of 50229.
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: 50229 / 79 = 635 with a remainder of 64.
Proof: (635 * 79) ... |
In a right-angled triangle, the lengths of the two legs are a = 3 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: 3^2 + 12^2 = c^2
2. Calculate the squares: 9 + 144 = c^2
3. Sum the squares: 153 = c^2
4. Take the square root of both sides: c = √153
Final Answ... |
Find the roots of the quadratic equation: 6x^2 + -10x + 7 = 0 | 1. Identify coefficients: a=6, b=-10, c=7.
2. Calculate the discriminant: Δ = b^2 - 4ac = -10^2 - 4(6)(7) = 100 - 168 = -68.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (10 ± √68i) / 12.
Final Answer: x = 0.8333333333333334 ± 0.6871842709362768i |
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 quotient and remainder of 77028 divided by 32. Show the long division steps. | To divide 77028 by 32:
1. Determine how many times 32 goes into the leading digits of 77028.
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: 77028 / 32 = 2407 with a remainder of 4.
Proof: (2407 * 32)... |
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) |
Find the roots of the quadratic equation: 10x^2 + -8x + -1 = 0 | 1. Identify coefficients: a=10, b=-8, c=-1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 64 - -40 = 104.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (8 ± √104) / 20.
5. Calculate roots: x1 = 0.9099019513592784, x2 = -0.10990195135927845.
Fina... |
Find the roots of the quadratic equation: 10x^2 + -1x + -3 = 0 | 1. Identify coefficients: a=10, b=-1, c=-3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1 - -120 = 121.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (1 ± √121) / 20.
5. Calculate roots: x1 = 0.6, x2 = -0.5.
Final Answer: x = 0.6, -0.5 |
In a right-angled triangle, the lengths of the two legs are a = 6 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: 6^2 + 12^2 = c^2
2. Calculate the squares: 36 + 144 = c^2
3. Sum the squares: 180 = c^2
4. Take the square root of both sides: c = √180
Final Ans... |
Compute the quotient and remainder of 75929 divided by 47. Show the long division steps. | To divide 75929 by 47:
1. Determine how many times 47 goes into the leading digits of 75929.
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: 75929 / 47 = 1615 with a remainder of 24.
Proof: (1615 * 47... |
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 14/6 and 14/8. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 6 and 8, which is 24.
2. Convert 14/6 by multiplying top and bottom by 4: 56/24.
3. Convert 14/8 by multiplying top and bottom by 3: 42/24.
4. Add the numerators: 56 + 42 = 98.
5. The resulting fraction is 98/24.
6. Simplify by d... |
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 |
Add the fractions 2/4 and 9/6. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 4 and 6, which is 12.
2. Convert 2/4 by multiplying top and bottom by 3: 6/12.
3. Convert 9/6 by multiplying top and bottom by 2: 18/12.
4. Add the numerators: 6 + 18 = 24.
5. The resulting fraction is 24/12.
6. Simplify by divid... |
Compute the quotient and remainder of 50755 divided by 96. Show the long division steps. | To divide 50755 by 96:
1. Determine how many times 96 goes into the leading digits of 50755.
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: 50755 / 96 = 528 with a remainder of 67.
Proof: (528 * 96) ... |
Compute the indefinite integral of f(x) = 10x^3. | To find the indefinite integral ∫ 10x^3 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 3 + 1 = 4.
2. Divide the coefficient by the new exponent: 10 / 4 = 5/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 10x^3 dx = 5/2x^4 + C |
In a right-angled triangle, the lengths of the two legs are a = 4 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: 4^2 + 11^2 = c^2
2. Calculate the squares: 16 + 121 = c^2
3. Sum the squares: 137 = c^2
4. Take the square root of both sides: c = √137
Final Ans... |
In a right-angled triangle, the lengths of the two legs are a = 4 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: 4^2 + 20^2 = c^2
2. Calculate the squares: 16 + 400 = c^2
3. Sum the squares: 416 = c^2
4. Take the square root of both sides: c = √416
Final Ans... |
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) |
Multiply the following 2x2 matrices:
A =
[5, 0]
[-3, -4]
B =
[-4, 3]
[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*-4) + (0*5) = -20
C12 = (A11 * B12) + (A12 * B22) = (5*3) + (0*4) = 15
C21 = (A21 * B11) + (A22 * B21) = (-3*-4) + (-4*5) = -8
C22 = (A21 * B12) + (A22 * B22) = (-3*3) + (-4*4) = -25
Resul... |
Compute the quotient and remainder of 41420 divided by 62. Show the long division steps. | To divide 41420 by 62:
1. Determine how many times 62 goes into the leading digits of 41420.
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: 41420 / 62 = 668 with a remainder of 4.
Proof: (668 * 62) +... |
Calculate the mean, population variance, and standard deviation for the dataset: [19, 8, 6, 5, 5] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (19 + 8 + 6 + 5 + 5) / 5 = 43 / 5 = 8.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [10.4, -0.6, -2.6, -3.6, -3.6]
Squared Differences: [108.16, 0.36, 6.76, 12.96, 12.96]
... |
Compute the quotient and remainder of 10436 divided by 12. Show the long division steps. | To divide 10436 by 12:
1. Determine how many times 12 goes into the leading digits of 10436.
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: 10436 / 12 = 869 with a remainder of 8.
Proof: (869 * 12) +... |
Calculate the mean, population variance, and standard deviation for the dataset: [10, 5, 20, 4, 17] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (10 + 5 + 20 + 4 + 17) / 5 = 56 / 5 = 11.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-1.2, -6.2, 8.8, -7.2, 5.8]
Squared Differences: [1.44, 38.44, 77.44, 51.84, 33.64]... |
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 |
Calculate the mean, population variance, and standard deviation for the dataset: [3, 4, 14, 7, 16] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (3 + 4 + 14 + 7 + 16) / 5 = 44 / 5 = 8.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-5.8, -4.8, 5.2, -1.8, 7.2]
Squared Differences: [33.64, 23.04, 27.04, 3.24, 51.84]
... |
Calculate the mean, population variance, and standard deviation for the dataset: [17, 5, 11, 2, 14] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (17 + 5 + 11 + 2 + 14) / 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, 1.2, -7.8, 4.2]
Squared Differences: [51.84, 23.04, 1.44, 60.84, 17.64]
... |
Evaluate sin(0°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(0°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.0000 (exact form depends on root simplification if applicable) |
Find the roots of the quadratic equation: 2x^2 + 6x + -1 = 0 | 1. Identify coefficients: a=2, b=6, c=-1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 36 - -8 = 44.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-6 ± √44) / 4.
5. Calculate roots: x1 = 0.1583123951776999, x2 = -3.1583123951777.
Final Answer:... |
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) |
In a right-angled triangle, the lengths of the two legs are a = 9 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: 9^2 + 19^2 = c^2
2. Calculate the squares: 81 + 361 = c^2
3. Sum the squares: 442 = c^2
4. Take the square root of both sides: c = √442
Final Ans... |
In a right-angled triangle, the lengths of the two legs are a = 5 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: 5^2 + 8^2 = c^2
2. Calculate the squares: 25 + 64 = c^2
3. Sum the squares: 89 = c^2
4. Take the square root of both sides: c = √89
Final Answer:... |
Multiply the following 2x2 matrices:
A =
[-2, -3]
[-4, 1]
B =
[-4, 3]
[1, -5] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-2*-4) + (-3*1) = 5
C12 = (A11 * B12) + (A12 * B22) = (-2*3) + (-3*-5) = 9
C21 = (A21 * B11) + (A22 * B21) = (-4*-4) + (1*1) = 17
C22 = (A21 * B12) + (A22 * B22) = (-4*3) + (1*-5) = -17
Resu... |
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