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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 ... |
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) |
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 +... |
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) |
Evaluate the logarithm: log_2(4) | To evaluate log_2(4), we ask the question: '2 raised to what power equals 4?'
Let x be the unknown power: 2^x = 4
Since 2^2 = 4, it follows that x = 2.
Final Answer: 2 |
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) |
In a right-angled triangle, the lengths of the two legs are a = 15 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: 15^2 + 17^2 = c^2
2. Calculate the squares: 225 + 289 = c^2
3. Sum the squares: 514 = c^2
4. Take the square root of both sides: c = √514
Final A... |
Evaluate the logarithm: log_2(32) | To evaluate log_2(32), we ask the question: '2 raised to what power equals 32?'
Let x be the unknown power: 2^x = 32
Since 2^5 = 32, it follows that x = 5.
Final Answer: 5 |
Calculate the mean, population variance, and standard deviation for the dataset: [13, 8, 7, 10, 5] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (13 + 8 + 7 + 10 + 5) / 5 = 43 / 5 = 8.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [4.4, -0.6, -1.6, 1.4, -3.6]
Squared Differences: [19.36, 0.36, 2.56, 1.96, 12.96]
... |
Find the roots of the quadratic equation: 1x^2 + 10x + 0 = 0 | 1. Identify coefficients: a=1, b=10, c=0.
2. Calculate the discriminant: Δ = b^2 - 4ac = 100 - 0 = 100.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-10 ± √100) / 2.
5. Calculate roots: x1 = 0.0, x2 = -10.0.
Final Answer: x = 0.0, -10.0 |
Compute the quotient and remainder of 33881 divided by 69. Show the long division steps. | To divide 33881 by 69:
1. Determine how many times 69 goes into the leading digits of 33881.
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: 33881 / 69 = 491 with a remainder of 2.
Proof: (491 * 69) +... |
Multiply the following 2x2 matrices:
A =
[-1, 3]
[-4, 0]
B =
[-4, 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*-4) + (3*-4) = -8
C12 = (A11 * B12) + (A12 * B22) = (-1*0) + (3*3) = 9
C21 = (A21 * B11) + (A22 * B21) = (-4*-4) + (0*-4) = 16
C22 = (A21 * B12) + (A22 * B22) = (-4*0) + (0*3) = 0
Result ... |
Evaluate the logarithm: log_3(243) | To evaluate log_3(243), we ask the question: '3 raised to what power equals 243?'
Let x be the unknown power: 3^x = 243
Since 3^5 = 243, it follows that x = 5.
Final Answer: 5 |
Find the roots of the quadratic equation: 8x^2 + -8x + -6 = 0 | 1. Identify coefficients: a=8, b=-8, c=-6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 64 - -192 = 256.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (8 ± √256) / 16.
5. Calculate roots: x1 = 1.5, x2 = -0.5.
Final Answer: x = 1.5, -0.5 |
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 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 ... |
Add the fractions 8/6 and 12/9. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 6 and 9, which is 18.
2. Convert 8/6 by multiplying top and bottom by 3: 24/18.
3. Convert 12/9 by multiplying top and bottom by 2: 24/18.
4. Add the numerators: 24 + 24 = 48.
5. The resulting fraction is 48/18.
6. Simplify by di... |
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 58911 divided by 45. Show the long division steps. | To divide 58911 by 45:
1. Determine how many times 45 goes into the leading digits of 58911.
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: 58911 / 45 = 1309 with a remainder of 6.
Proof: (1309 * 45)... |
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 |
In a right-angled triangle, the lengths of the two legs are a = 6 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: 6^2 + 13^2 = c^2
2. Calculate the squares: 36 + 169 = c^2
3. Sum the squares: 205 = c^2
4. Take the square root of both sides: c = √205
Final Ans... |
Multiply the following 2x2 matrices:
A =
[-2, -2]
[-1, -3]
B =
[0, 4]
[-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) = (-2*0) + (-2*-4) = 8
C12 = (A11 * B12) + (A12 * B22) = (-2*4) + (-2*-4) = 0
C21 = (A21 * B11) + (A22 * B21) = (-1*0) + (-3*-4) = 12
C22 = (A21 * B12) + (A22 * B22) = (-1*4) + (-3*-4) = 8
Resu... |
Compute the quotient and remainder of 43597 divided by 51. Show the long division steps. | To divide 43597 by 51:
1. Determine how many times 51 goes into the leading digits of 43597.
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: 43597 / 51 = 854 with a remainder of 43.
Proof: (854 * 51) ... |
Compute the quotient and remainder of 43062 divided by 59. Show the long division steps. | To divide 43062 by 59:
1. Determine how many times 59 goes into the leading digits of 43062.
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: 43062 / 59 = 729 with a remainder of 51.
Proof: (729 * 59) ... |
Multiply the following 2x2 matrices:
A =
[0, 1]
[-3, 0]
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) = (0*-3) + (1*-2) = -2
C12 = (A11 * B12) + (A12 * B22) = (0*-4) + (1*-1) = -1
C21 = (A21 * B11) + (A22 * B21) = (-3*-3) + (0*-2) = 9
C22 = (A21 * B12) + (A22 * B22) = (-3*-4) + (0*-1) = 12
Resu... |
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 |
Find the roots of the quadratic equation: 8x^2 + 0x + 6 = 0 | 1. Identify coefficients: a=8, b=0, c=6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 0^2 - 4(8)(6) = 0 - 192 = -192.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (0 ± √192i) / 16.
Final Answer: x = 0.0 ± 0.8660254037844386i |
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) |
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 |
Add the fractions 3/7 and 3/9. 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 3/7 by multiplying top and bottom by 9: 27/63.
3. Convert 3/9 by multiplying top and bottom by 7: 21/63.
4. Add the numerators: 27 + 21 = 48.
5. The resulting fraction is 48/63.
6. Simplify by div... |
Compute the indefinite integral of f(x) = 1x^1. | To find the indefinite integral ∫ 1x^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: 1 / 2 = 1/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 1x^1 dx = 1/2x^2 + C |
Multiply the following 2x2 matrices:
A =
[-4, 3]
[-2, -2]
B =
[3, -3]
[-3, 1] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-4*3) + (3*-3) = -21
C12 = (A11 * B12) + (A12 * B22) = (-4*-3) + (3*1) = 15
C21 = (A21 * B11) + (A22 * B21) = (-2*3) + (-2*-3) = 0
C22 = (A21 * B12) + (A22 * B22) = (-2*-3) + (-2*1) = 4
Resu... |
Compute the quotient and remainder of 43330 divided by 74. Show the long division steps. | To divide 43330 by 74:
1. Determine how many times 74 goes into the leading digits of 43330.
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: 43330 / 74 = 585 with a remainder of 40.
Proof: (585 * 74) ... |
Compute the indefinite integral of f(x) = 4x^3. | To find the indefinite integral ∫ 4x^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: 4 / 4 = 1.
3. Combine and add the constant of integration C.
Final Answer: ∫ 4x^3 dx = 1x^4 + C |
Compute the quotient and remainder of 17158 divided by 68. Show the long division steps. | To divide 17158 by 68:
1. Determine how many times 68 goes into the leading digits of 17158.
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: 17158 / 68 = 252 with a remainder of 22.
Proof: (252 * 68) ... |
Compute the quotient and remainder of 68027 divided by 73. Show the long division steps. | To divide 68027 by 73:
1. Determine how many times 73 goes into the leading digits of 68027.
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: 68027 / 73 = 931 with a remainder of 64.
Proof: (931 * 73) ... |
Multiply the following 2x2 matrices:
A =
[-5, 0]
[-2, -5]
B =
[-3, 3]
[-3, -5] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-5*-3) + (0*-3) = 15
C12 = (A11 * B12) + (A12 * B22) = (-5*3) + (0*-5) = -15
C21 = (A21 * B11) + (A22 * B21) = (-2*-3) + (-5*-3) = 21
C22 = (A21 * B12) + (A22 * B22) = (-2*3) + (-5*-5) = 19
... |
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) |
In a right-angled triangle, the lengths of the two legs are a = 16 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: 16^2 + 16^2 = c^2
2. Calculate the squares: 256 + 256 = c^2
3. Sum the squares: 512 = c^2
4. Take the square root of both sides: c = √512
Final A... |
Multiply the following 2x2 matrices:
A =
[-3, 4]
[5, 2]
B =
[-3, -3]
[-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*-3) + (4*-5) = -11
C12 = (A11 * B12) + (A12 * B22) = (-3*-3) + (4*-1) = 5
C21 = (A21 * B11) + (A22 * B21) = (5*-3) + (2*-5) = -25
C22 = (A21 * B12) + (A22 * B22) = (5*-3) + (2*-1) = -17
R... |
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 |
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 +... |
Add the fractions 6/5 and 15/9. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 5 and 9, which is 45.
2. Convert 6/5 by multiplying top and bottom by 9: 54/45.
3. Convert 15/9 by multiplying top and bottom by 5: 75/45.
4. Add the numerators: 54 + 75 = 129.
5. The resulting fraction is 129/45.
6. Simplify by ... |
Find the roots of the quadratic equation: 7x^2 + 5x + 3 = 0 | 1. Identify coefficients: a=7, b=5, c=3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 5^2 - 4(7)(3) = 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) / 14.
Final Answer: x = -0.35714285714285715 ± 0.5486532677049005i |
Compute the quotient and remainder of 40045 divided by 53. Show the long division steps. | To divide 40045 by 53:
1. Determine how many times 53 goes into the leading digits of 40045.
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: 40045 / 53 = 755 with a remainder of 30.
Proof: (755 * 53) ... |
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 ... |
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 |
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 = 19 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: 19^2 + 4^2 = c^2
2. Calculate the squares: 361 + 16 = c^2
3. Sum the squares: 377 = c^2
4. Take the square root of both sides: c = √377
Final Ans... |
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 +... |
Find the roots of the quadratic equation: 2x^2 + 5x + -10 = 0 | 1. Identify coefficients: a=2, b=5, c=-10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 25 - -80 = 105.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-5 ± √105) / 4.
5. Calculate roots: x1 = 1.3117376914898995, x2 = -3.8117376914898995.
Final ... |
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(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) |
Multiply the following 2x2 matrices:
A =
[-4, 3]
[-4, 5]
B =
[2, 0]
[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) = (-4*2) + (3*5) = 7
C12 = (A11 * B12) + (A12 * B22) = (-4*0) + (3*1) = 3
C21 = (A21 * B11) + (A22 * B21) = (-4*2) + (5*5) = 17
C22 = (A21 * B12) + (A22 * B22) = (-4*0) + (5*1) = 5
Result Matri... |
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 |
Find the roots of the quadratic equation: 4x^2 + -6x + 9 = 0 | 1. Identify coefficients: a=4, b=-6, c=9.
2. Calculate the discriminant: Δ = b^2 - 4ac = -6^2 - 4(4)(9) = 36 - 144 = -108.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (6 ± √108i) / 8.
Final Answer: x = 0.75 ± 1.299038105676658i |
Compute the quotient and remainder of 34162 divided by 73. Show the long division steps. | To divide 34162 by 73:
1. Determine how many times 73 goes into the leading digits of 34162.
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: 34162 / 73 = 467 with a remainder of 71.
Proof: (467 * 73) ... |
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) |
In a right-angled triangle, the lengths of the two legs are a = 5 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: 5^2 + 10^2 = c^2
2. Calculate the squares: 25 + 100 = c^2
3. Sum the squares: 125 = c^2
4. Take the square root of both sides: c = √125
Final Ans... |
Add the fractions 11/12 and 12/4. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 12 and 4, which is 12.
2. Convert 11/12 by multiplying top and bottom by 1: 11/12.
3. Convert 12/4 by multiplying top and bottom by 3: 36/12.
4. Add the numerators: 11 + 36 = 47.
5. The resulting fraction is 47/12.
Final Answer: ... |
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 |
Find the roots of the quadratic equation: 4x^2 + 6x + -7 = 0 | 1. Identify coefficients: a=4, b=6, c=-7.
2. Calculate the discriminant: Δ = b^2 - 4ac = 36 - -112 = 148.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-6 ± √148) / 8.
5. Calculate roots: x1 = 0.7706906325745548, x2 = -2.2706906325745546.
Final ... |
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 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 quotient and remainder of 43682 divided by 84. Show the long division steps. | To divide 43682 by 84:
1. Determine how many times 84 goes into the leading digits of 43682.
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: 43682 / 84 = 520 with a remainder of 2.
Proof: (520 * 84) +... |
Add the fractions 13/8 and 3/6. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 8 and 6, which is 24.
2. Convert 13/8 by multiplying top and bottom by 3: 39/24.
3. Convert 3/6 by multiplying top and bottom by 4: 12/24.
4. Add the numerators: 39 + 12 = 51.
5. The resulting fraction is 51/24.
6. Simplify by di... |
Add the fractions 15/4 and 11/6. 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 15/4 by multiplying top and bottom by 3: 45/12.
3. Convert 11/6 by multiplying top and bottom by 2: 22/12.
4. Add the numerators: 45 + 22 = 67.
5. The resulting fraction is 67/12.
Final Answer: 67... |
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 ... |
Add the fractions 15/6 and 9/14. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 6 and 14, which is 42.
2. Convert 15/6 by multiplying top and bottom by 7: 105/42.
3. Convert 9/14 by multiplying top and bottom by 3: 27/42.
4. Add the numerators: 105 + 27 = 132.
5. The resulting fraction is 132/42.
6. Simplify... |
Find the derivative of f(x) = (5x + 4)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 5x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 5x + 4. Its derivative is h'(x) = 5.
4. Multiply g'(u) by h'(x): f'(x) = 3(5x + 4)^2 * 5 = 15(5x ... |
Add the fractions 3/2 and 13/12. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 2 and 12, which is 12.
2. Convert 3/2 by multiplying top and bottom by 6: 18/12.
3. Convert 13/12 by multiplying top and bottom by 1: 13/12.
4. Add the numerators: 18 + 13 = 31.
5. The resulting fraction is 31/12.
Final Answer: 3... |
In a right-angled triangle, the lengths of the two legs are a = 4 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: 4^2 + 12^2 = c^2
2. Calculate the squares: 16 + 144 = c^2
3. Sum the squares: 160 = c^2
4. Take the square root of both sides: c = √160
Final Ans... |
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 |
Multiply the following 2x2 matrices:
A =
[-3, 2]
[-2, -5]
B =
[-5, 1]
[-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) = (-3*-5) + (2*-2) = 11
C12 = (A11 * B12) + (A12 * B22) = (-3*1) + (2*0) = -3
C21 = (A21 * B11) + (A22 * B21) = (-2*-5) + (-5*-2) = 20
C22 = (A21 * B12) + (A22 * B22) = (-2*1) + (-5*0) = -2
Res... |
Compute the quotient and remainder of 7314 divided by 20. Show the long division steps. | To divide 7314 by 20:
1. Determine how many times 20 goes into the leading digits of 7314.
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: 7314 / 20 = 365 with a remainder of 14.
Proof: (365 * 20) + 1... |
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 +... |
In a right-angled triangle, the lengths of the two legs are a = 7 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: 7^2 + 4^2 = c^2
2. Calculate the squares: 49 + 16 = c^2
3. Sum the squares: 65 = c^2
4. Take the square root of both sides: c = √65
Final Answer:... |
Compute the quotient and remainder of 59885 divided by 54. Show the long division steps. | To divide 59885 by 54:
1. Determine how many times 54 goes into the leading digits of 59885.
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: 59885 / 54 = 1108 with a remainder of 53.
Proof: (1108 * 54... |
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 13/4 and 5/8. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 4 and 8, which is 8.
2. Convert 13/4 by multiplying top and bottom by 2: 26/8.
3. Convert 5/8 by multiplying top and bottom by 1: 5/8.
4. Add the numerators: 26 + 5 = 31.
5. The resulting fraction is 31/8.
Final Answer: 31/8 |
Find the roots of the quadratic equation: 6x^2 + -10x + -6 = 0 | 1. Identify coefficients: a=6, b=-10, c=-6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 100 - -144 = 244.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (10 ± √244) / 12.
5. Calculate roots: x1 = 2.135041612651109, x2 = -0.46837494598444235.
Fi... |
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) |
In a right-angled triangle, the lengths of the two legs are a = 8 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: 8^2 + 8^2 = c^2
2. Calculate the squares: 64 + 64 = c^2
3. Sum the squares: 128 = c^2
4. Take the square root of both sides: c = √128
Final Answe... |
Compute the quotient and remainder of 67213 divided by 91. Show the long division steps. | To divide 67213 by 91:
1. Determine how many times 91 goes into the leading digits of 67213.
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: 67213 / 91 = 738 with a remainder of 55.
Proof: (738 * 91) ... |
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 quotient and remainder of 20875 divided by 79. Show the long division steps. | To divide 20875 by 79:
1. Determine how many times 79 goes into the leading digits of 20875.
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: 20875 / 79 = 264 with a remainder of 19.
Proof: (264 * 79) ... |
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 |
Compute the quotient and remainder of 57946 divided by 80. Show the long division steps. | To divide 57946 by 80:
1. Determine how many times 80 goes into the leading digits of 57946.
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: 57946 / 80 = 724 with a remainder of 26.
Proof: (724 * 80) ... |
Calculate the mean, population variance, and standard deviation for the dataset: [19, 20, 3, 13, 1] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (19 + 20 + 3 + 13 + 1) / 5 = 56 / 5 = 11.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [7.8, 8.8, -8.2, 1.8, -10.2]
Squared Differences: [60.84, 77.44, 67.24, 3.24, 104.04... |
Multiply the following 2x2 matrices:
A =
[3, -3]
[-4, 1]
B =
[-1, 1]
[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) = (3*-1) + (-3*2) = -9
C12 = (A11 * B12) + (A12 * B22) = (3*1) + (-3*0) = 3
C21 = (A21 * B11) + (A22 * B21) = (-4*-1) + (1*2) = 6
C22 = (A21 * B12) + (A22 * B22) = (-4*1) + (1*0) = -4
Result Ma... |
Add the fractions 1/15 and 15/6. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 15 and 6, which is 30.
2. Convert 1/15 by multiplying top and bottom by 2: 2/30.
3. Convert 15/6 by multiplying top and bottom by 5: 75/30.
4. Add the numerators: 2 + 75 = 77.
5. The resulting fraction is 77/30.
Final Answer: 77/... |
Multiply the following 2x2 matrices:
A =
[-4, 3]
[-3, -2]
B =
[-5, -5]
[4, 1] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-4*-5) + (3*4) = 32
C12 = (A11 * B12) + (A12 * B22) = (-4*-5) + (3*1) = 23
C21 = (A21 * B11) + (A22 * B21) = (-3*-5) + (-2*4) = 7
C22 = (A21 * B12) + (A22 * B22) = (-3*-5) + (-2*1) = 13
Resu... |
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) |
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 63172 divided by 90. Show the long division steps. | To divide 63172 by 90:
1. Determine how many times 90 goes into the leading digits of 63172.
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: 63172 / 90 = 701 with a remainder of 82.
Proof: (701 * 90) ... |
Add the fractions 14/2 and 5/14. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 2 and 14, which is 14.
2. Convert 14/2 by multiplying top and bottom by 7: 98/14.
3. Convert 5/14 by multiplying top and bottom by 1: 5/14.
4. Add the numerators: 98 + 5 = 103.
5. The resulting fraction is 103/14.
Final Answer: 1... |
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) = 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: 1x^2 + -3x + -9 = 0 | 1. Identify coefficients: a=1, b=-3, c=-9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 9 - -36 = 45.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (3 ± √45) / 2.
5. Calculate roots: x1 = 4.854101966249685, x2 = -1.8541019662496847.
Final Answe... |
Find the derivative of f(x) = (4x + 4)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 4x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 4x + 4. Its derivative is h'(x) = 4.
4. Multiply g'(u) by h'(x): f'(x) = 3(4x + 4)^2 * 4 = 12(4x ... |
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