prompt stringlengths 32 115 | completion stringlengths 167 566 |
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Multiply the following 2x2 matrices:
A =
[5, -1]
[-4, -5]
B =
[1, 2]
[-4, 3] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (5*1) + (-1*-4) = 9
C12 = (A11 * B12) + (A12 * B22) = (5*2) + (-1*3) = 7
C21 = (A21 * B11) + (A22 * B21) = (-4*1) + (-5*-4) = 16
C22 = (A21 * B12) + (A22 * B22) = (-4*2) + (-5*3) = -23
Result... |
Multiply the following 2x2 matrices:
A =
[3, -2]
[0, 2]
B =
[-4, 0]
[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) = (3*-4) + (-2*1) = -14
C12 = (A11 * B12) + (A12 * B22) = (3*0) + (-2*-5) = 10
C21 = (A21 * B11) + (A22 * B21) = (0*-4) + (2*1) = 2
C22 = (A21 * B12) + (A22 * B22) = (0*0) + (2*-5) = -10
Result... |
Compute the quotient and remainder of 47956 divided by 92. Show the long division steps. | To divide 47956 by 92:
1. Determine how many times 92 goes into the leading digits of 47956.
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: 47956 / 92 = 521 with a remainder of 24.
Proof: (521 * 92) ... |
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) |
Multiply the following 2x2 matrices:
A =
[-1, 5]
[4, -3]
B =
[2, 1]
[4, 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*2) + (5*4) = 18
C12 = (A11 * B12) + (A12 * B22) = (-1*1) + (5*0) = -1
C21 = (A21 * B11) + (A22 * B21) = (4*2) + (-3*4) = -4
C22 = (A21 * B12) + (A22 * B22) = (4*1) + (-3*0) = 4
Result Mat... |
Find the roots of the quadratic equation: 5x^2 + -2x + 9 = 0 | 1. Identify coefficients: a=5, b=-2, c=9.
2. Calculate the discriminant: Δ = b^2 - 4ac = -2^2 - 4(5)(9) = 4 - 180 = -176.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (2 ± √176i) / 10.
Final Answer: x = 0.2 ± 1.3266499161421599i |
Find the roots of the quadratic equation: 3x^2 + 4x + 9 = 0 | 1. Identify coefficients: a=3, b=4, c=9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 4^2 - 4(3)(9) = 16 - 108 = -92.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-4 ± √92i) / 6.
Final Answer: x = -0.6666666666666666 ± 1.5986105077709063i |
Compute the indefinite integral of f(x) = 6x^5. | To find the indefinite integral ∫ 6x^5 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 5 + 1 = 6.
2. Divide the coefficient by the new exponent: 6 / 6 = 1.
3. Combine and add the constant of integration C.
Final Answer: ∫ 6x^5 dx = 1x^6 + C |
Compute the quotient and remainder of 45636 divided by 19. Show the long division steps. | To divide 45636 by 19:
1. Determine how many times 19 goes into the leading digits of 45636.
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: 45636 / 19 = 2401 with a remainder of 17.
Proof: (2401 * 19... |
Multiply the following 2x2 matrices:
A =
[2, -1]
[-2, -2]
B =
[4, -5]
[2, 1] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (2*4) + (-1*2) = 6
C12 = (A11 * B12) + (A12 * B22) = (2*-5) + (-1*1) = -11
C21 = (A21 * B11) + (A22 * B21) = (-2*4) + (-2*2) = -12
C22 = (A21 * B12) + (A22 * B22) = (-2*-5) + (-2*1) = 8
Resul... |
Multiply the following 2x2 matrices:
A =
[-2, -4]
[-3, 2]
B =
[1, -2]
[-1, -4] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-2*1) + (-4*-1) = 2
C12 = (A11 * B12) + (A12 * B22) = (-2*-2) + (-4*-4) = 20
C21 = (A21 * B11) + (A22 * B21) = (-3*1) + (2*-1) = -5
C22 = (A21 * B12) + (A22 * B22) = (-3*-2) + (2*-4) = -2
Re... |
Find the derivative of f(x) = (3x + 2)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 3x + 2.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 2. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 2)^2 * 3 = 9(3x +... |
Find the derivative of f(x) = (2x + 4)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 2x + 4.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 2x + 4. Its derivative is h'(x) = 2.
4. Multiply g'(u) by h'(x): f'(x) = 3(2x + 4)^2 * 2 = 6(2x +... |
Find the 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 54624 divided by 2. Show the long division steps. | To divide 54624 by 2:
1. Determine how many times 2 goes into the leading digits of 54624.
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: 54624 / 2 = 27312 with a remainder of 0.
Proof: (27312 * 2) +... |
Compute the indefinite integral of f(x) = 8x^1. | To find the indefinite integral ∫ 8x^1 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 1 + 1 = 2.
2. Divide the coefficient by the new exponent: 8 / 2 = 4.
3. Combine and add the constant of integration C.
Final Answer: ∫ 8x^1 dx = 4x^2 + C |
Add the fractions 6/5 and 10/3. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 5 and 3, which is 15.
2. Convert 6/5 by multiplying top and bottom by 3: 18/15.
3. Convert 10/3 by multiplying top and bottom by 5: 50/15.
4. Add the numerators: 18 + 50 = 68.
5. The resulting fraction is 68/15.
Final Answer: 68/... |
Evaluate tan(30°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(30°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.5774 (exact form depends on root simplification if applicable) |
Find the 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 ... |
Calculate the mean, population variance, and standard deviation for the dataset: [14, 10, 19, 9, 1] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (14 + 10 + 19 + 9 + 1) / 5 = 53 / 5 = 10.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [3.4, -0.6, 8.4, -1.6, -9.6]
Squared Differences: [11.56, 0.36, 70.56, 2.56, 92.16]
... |
Add the fractions 4/12 and 3/8. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 12 and 8, which is 24.
2. Convert 4/12 by multiplying top and bottom by 2: 8/24.
3. Convert 3/8 by multiplying top and bottom by 3: 9/24.
4. Add the numerators: 8 + 9 = 17.
5. The resulting fraction is 17/24.
Final Answer: 17/24 |
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 41228 divided by 2. Show the long division steps. | To divide 41228 by 2:
1. Determine how many times 2 goes into the leading digits of 41228.
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: 41228 / 2 = 20614 with a remainder of 0.
Proof: (20614 * 2) +... |
Multiply the following 2x2 matrices:
A =
[-5, 0]
[-2, -1]
B =
[0, -3]
[-4, 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*0) + (0*-4) = 0
C12 = (A11 * B12) + (A12 * B22) = (-5*-3) + (0*5) = 15
C21 = (A21 * B11) + (A22 * B21) = (-2*0) + (-1*-4) = 4
C22 = (A21 * B12) + (A22 * B22) = (-2*-3) + (-1*5) = 1
Result... |
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 |
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) |
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 quotient and remainder of 99229 divided by 29. Show the long division steps. | To divide 99229 by 29:
1. Determine how many times 29 goes into the leading digits of 99229.
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: 99229 / 29 = 3421 with a remainder of 20.
Proof: (3421 * 29... |
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: 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 = 1.8541019662496847, x2 = -4.854101966249685.
Final Answe... |
Multiply the following 2x2 matrices:
A =
[-5, 5]
[-1, -2]
B =
[4, 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*4) + (5*3) = -5
C12 = (A11 * B12) + (A12 * B22) = (-5*3) + (5*-5) = -40
C21 = (A21 * B11) + (A22 * B21) = (-1*4) + (-2*3) = -10
C22 = (A21 * B12) + (A22 * B22) = (-1*3) + (-2*-5) = 7
Resu... |
Find the roots of the quadratic equation: 5x^2 + 0x + -9 = 0 | 1. Identify coefficients: a=5, b=0, c=-9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 0 - -180 = 180.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (0 ± √180) / 10.
5. Calculate roots: x1 = 1.3416407864998738, x2 = -1.3416407864998738.
Final A... |
In a right-angled triangle, the lengths of the two legs are a = 10 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: 10^2 + 11^2 = c^2
2. Calculate the squares: 100 + 121 = c^2
3. Sum the squares: 221 = c^2
4. Take the square root of both sides: c = √221
Final A... |
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: [8, 11, 2, 7, 6] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (8 + 11 + 2 + 7 + 6) / 5 = 34 / 5 = 6.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [1.2, 4.2, -4.8, 0.2, -0.8]
Squared Differences: [1.44, 17.64, 23.04, 0.04, 0.64]
Su... |
Compute the quotient and remainder of 70076 divided by 13. Show the long division steps. | To divide 70076 by 13:
1. Determine how many times 13 goes into the leading digits of 70076.
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: 70076 / 13 = 5390 with a remainder of 6.
Proof: (5390 * 13)... |
Compute the quotient and remainder of 38592 divided by 60. Show the long division steps. | To divide 38592 by 60:
1. Determine how many times 60 goes into the leading digits of 38592.
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: 38592 / 60 = 643 with a remainder of 12.
Proof: (643 * 60) ... |
Find the roots of the quadratic equation: 3x^2 + 1x + -3 = 0 | 1. Identify coefficients: a=3, b=1, c=-3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 1 - -36 = 37.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-1 ± √37) / 6.
5. Calculate roots: x1 = 0.8471270883830365, x2 = -1.18046042171637.
Final Answer... |
Multiply the following 2x2 matrices:
A =
[-1, 4]
[0, 5]
B =
[2, -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) = (-1*2) + (4*-3) = -14
C12 = (A11 * B12) + (A12 * B22) = (-1*-3) + (4*2) = 11
C21 = (A21 * B11) + (A22 * B21) = (0*2) + (5*-3) = -15
C22 = (A21 * B12) + (A22 * B22) = (0*-3) + (5*2) = 10
Resul... |
Compute the quotient and remainder of 60645 divided by 44. Show the long division steps. | To divide 60645 by 44:
1. Determine how many times 44 goes into the leading digits of 60645.
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: 60645 / 44 = 1378 with a remainder of 13.
Proof: (1378 * 44... |
Compute the indefinite integral of f(x) = 4x^5. | To find the indefinite integral ∫ 4x^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: 4 / 6 = 2/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 4x^5 dx = 2/3x^6 + C |
Calculate the mean, population variance, and standard deviation for the dataset: [1, 4, 17, 15, 19] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (1 + 4 + 17 + 15 + 19) / 5 = 56 / 5 = 11.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-10.2, -7.2, 5.8, 3.8, 7.8]
Squared Differences: [104.04, 51.84, 33.64, 14.44, 60.8... |
Find the roots of the quadratic equation: 4x^2 + 10x + 1 = 0 | 1. Identify coefficients: a=4, b=10, c=1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 100 - 16 = 84.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-10 ± √84) / 8.
5. Calculate roots: x1 = -0.10435607626104004, x2 = -2.3956439237389597.
Final ... |
Evaluate tan(0°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of tan(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) |
In a right-angled triangle, the lengths of the two legs are a = 18 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: 18^2 + 12^2 = c^2
2. Calculate the squares: 324 + 144 = c^2
3. Sum the squares: 468 = c^2
4. Take the square root of both sides: c = √468
Final A... |
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 |
Compute the indefinite integral of f(x) = 10x^1. | To find the indefinite integral ∫ 10x^1 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 1 + 1 = 2.
2. Divide the coefficient by the new exponent: 10 / 2 = 5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 10x^1 dx = 5x^2 + C |
In a right-angled triangle, the lengths of the two legs are a = 19 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: 19^2 + 3^2 = c^2
2. Calculate the squares: 361 + 9 = c^2
3. Sum the squares: 370 = c^2
4. Take the square root of both sides: c = √370
Final Answ... |
Add the fractions 1/3 and 11/12. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 3 and 12, which is 12.
2. Convert 1/3 by multiplying top and bottom by 4: 4/12.
3. Convert 11/12 by multiplying top and bottom by 1: 11/12.
4. Add the numerators: 4 + 11 = 15.
5. The resulting fraction is 15/12.
6. Simplify by di... |
Multiply the following 2x2 matrices:
A =
[-5, -1]
[-4, 5]
B =
[2, -2]
[0, 3] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (-5*2) + (-1*0) = -10
C12 = (A11 * B12) + (A12 * B22) = (-5*-2) + (-1*3) = 7
C21 = (A21 * B11) + (A22 * B21) = (-4*2) + (5*0) = -8
C22 = (A21 * B12) + (A22 * B22) = (-4*-2) + (5*3) = 23
Resul... |
Find the roots of the quadratic equation: 5x^2 + 0x + -3 = 0 | 1. Identify coefficients: a=5, b=0, c=-3.
2. Calculate the discriminant: Δ = b^2 - 4ac = 0 - -60 = 60.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (0 ± √60) / 10.
5. Calculate roots: x1 = 0.7745966692414834, x2 = -0.7745966692414834.
Final Answ... |
Add the fractions 3/2 and 14/5. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 2 and 5, which is 10.
2. Convert 3/2 by multiplying top and bottom by 5: 15/10.
3. Convert 14/5 by multiplying top and bottom by 2: 28/10.
4. Add the numerators: 15 + 28 = 43.
5. The resulting fraction is 43/10.
Final Answer: 43/... |
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 69573 divided by 37. Show the long division steps. | To divide 69573 by 37:
1. Determine how many times 37 goes into the leading digits of 69573.
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: 69573 / 37 = 1880 with a remainder of 13.
Proof: (1880 * 37... |
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) = 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 8/11 and 3/12. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 11 and 12, which is 132.
2. Convert 8/11 by multiplying top and bottom by 12: 96/132.
3. Convert 3/12 by multiplying top and bottom by 11: 33/132.
4. Add the numerators: 96 + 33 = 129.
5. The resulting fraction is 129/132.
6. Sim... |
Compute the indefinite integral of f(x) = 7x^4. | To find the indefinite integral ∫ 7x^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: 7 / 5 = 7/5.
3. Combine and add the constant of integration C.
Final Answer: ∫ 7x^4 dx = 7/5x^5 + C |
Calculate the mean, population variance, and standard deviation for the dataset: [8, 5, 14, 15, 16] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (8 + 5 + 14 + 15 + 16) / 5 = 58 / 5 = 11.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-3.6, -6.6, 2.4, 3.4, 4.4]
Squared Differences: [12.96, 43.56, 5.76, 11.56, 19.36]
... |
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: 6x^2 + 3x + -6 = 0 | 1. Identify coefficients: a=6, b=3, c=-6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 9 - -144 = 153.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-3 ± √153) / 12.
5. Calculate roots: x1 = 0.780776406404415, x2 = -1.2807764064044151.
Final A... |
Multiply the following 2x2 matrices:
A =
[0, 4]
[-3, -3]
B =
[4, -2]
[3, 4] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (0*4) + (4*3) = 12
C12 = (A11 * B12) + (A12 * B22) = (0*-2) + (4*4) = 16
C21 = (A21 * B11) + (A22 * B21) = (-3*4) + (-3*3) = -21
C22 = (A21 * B12) + (A22 * B22) = (-3*-2) + (-3*4) = -6
Result... |
Compute the quotient and remainder of 95795 divided by 17. Show the long division steps. | To divide 95795 by 17:
1. Determine how many times 17 goes into the leading digits of 95795.
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: 95795 / 17 = 5635 with a remainder of 0.
Proof: (5635 * 17)... |
Multiply the following 2x2 matrices:
A =
[4, 0]
[0, 5]
B =
[-5, -5]
[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) = (4*-5) + (0*5) = -20
C12 = (A11 * B12) + (A12 * B22) = (4*-5) + (0*2) = -20
C21 = (A21 * B11) + (A22 * B21) = (0*-5) + (5*5) = 25
C22 = (A21 * B12) + (A22 * B22) = (0*-5) + (5*2) = 10
Result ... |
Compute the quotient and remainder of 76799 divided by 36. Show the long division steps. | To divide 76799 by 36:
1. Determine how many times 36 goes into the leading digits of 76799.
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: 76799 / 36 = 2133 with a remainder of 11.
Proof: (2133 * 36... |
Add the fractions 13/4 and 1/14. 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 13/4 by multiplying top and bottom by 7: 91/28.
3. Convert 1/14 by multiplying top and bottom by 2: 2/28.
4. Add the numerators: 91 + 2 = 93.
5. The resulting fraction is 93/28.
Final Answer: 93/... |
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 ... |
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 +... |
Compute the quotient and remainder of 48117 divided by 11. Show the long division steps. | To divide 48117 by 11:
1. Determine how many times 11 goes into the leading digits of 48117.
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: 48117 / 11 = 4374 with a remainder of 3.
Proof: (4374 * 11)... |
Evaluate cos(90°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of cos(90°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.0000 (exact form depends on root simplification if applicable) |
Evaluate the logarithm: log_5(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: 10x^2 + -8x + -4 = 0 | 1. Identify coefficients: a=10, b=-8, c=-4.
2. Calculate the discriminant: Δ = b^2 - 4ac = 64 - -160 = 224.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (8 ± √224) / 20.
5. Calculate roots: x1 = 1.1483314773547881, x2 = -0.34833147735478825.
Fin... |
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) |
Calculate the mean, population variance, and standard deviation for the dataset: [11, 9, 13, 6, 19] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (11 + 9 + 13 + 6 + 19) / 5 = 58 / 5 = 11.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-0.6, -2.6, 1.4, -5.6, 7.4]
Squared Differences: [0.36, 6.76, 1.96, 31.36, 54.76]
... |
Add the fractions 1/2 and 1/11. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 2 and 11, which is 22.
2. Convert 1/2 by multiplying top and bottom by 11: 11/22.
3. Convert 1/11 by multiplying top and bottom by 2: 2/22.
4. Add the numerators: 11 + 2 = 13.
5. The resulting fraction is 13/22.
Final Answer: 13/... |
Add the fractions 7/6 and 12/7. 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 7/6 by multiplying top and bottom by 7: 49/42.
3. Convert 12/7 by multiplying top and bottom by 6: 72/42.
4. Add the numerators: 49 + 72 = 121.
5. The resulting fraction is 121/42.
Final Answer: 1... |
Compute the quotient and remainder of 35498 divided by 67. Show the long division steps. | To divide 35498 by 67:
1. Determine how many times 67 goes into the leading digits of 35498.
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: 35498 / 67 = 529 with a remainder of 55.
Proof: (529 * 67) ... |
Find the roots of the quadratic equation: 5x^2 + -4x + -8 = 0 | 1. Identify coefficients: a=5, b=-4, c=-8.
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) / 10.
5. Calculate roots: x1 = 1.72664991614216, x2 = -0.92664991614216.
Final Ans... |
In a right-angled triangle, the lengths of the two legs are a = 13 and b = 18. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 13^2 + 18^2 = c^2
2. Calculate the squares: 169 + 324 = c^2
3. Sum the squares: 493 = c^2
4. Take the square root of both sides: c = √493
Final A... |
Find the derivative of f(x) = (3x + 3)^3 using the chain rule. | The chain rule states that if f(x) = g(h(x)), then f'(x) = g'(h(x)) * h'(x).
1. Let the outer function g(u) = u^3, where u = 3x + 3.
2. The derivative of the outer function is g'(u) = 3u^2.
3. Let the inner function h(x) = 3x + 3. Its derivative is h'(x) = 3.
4. Multiply g'(u) by h'(x): f'(x) = 3(3x + 3)^2 * 3 = 9(3x +... |
Multiply the following 2x2 matrices:
A =
[5, 2]
[3, 0]
B =
[5, -5]
[3, -3] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (5*5) + (2*3) = 31
C12 = (A11 * B12) + (A12 * B22) = (5*-5) + (2*-3) = -31
C21 = (A21 * B11) + (A22 * B21) = (3*5) + (0*3) = 15
C22 = (A21 * B12) + (A22 * B22) = (3*-5) + (0*-3) = -15
Result ... |
Find the roots of the quadratic equation: 8x^2 + 6x + 9 = 0 | 1. Identify coefficients: a=8, b=6, c=9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 6^2 - 4(8)(9) = 36 - 288 = -252.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-6 ± √252i) / 16.
Final Answer: x = -0.375 ± 0.9921567416492215i |
Calculate the mean, population variance, and standard deviation for the dataset: [9, 20, 20, 19, 18] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (9 + 20 + 20 + 19 + 18) / 5 = 86 / 5 = 17.2
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-8.2, 2.8, 2.8, 1.8, 0.8]
Squared Differences: [67.24, 7.84, 7.84, 3.24, 0.64]
... |
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 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 roots of the quadratic equation: 8x^2 + 2x + 9 = 0 | 1. Identify coefficients: a=8, b=2, c=9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 2^2 - 4(8)(9) = 4 - 288 = -284.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-2 ± √284i) / 16.
Final Answer: x = -0.125 ± 1.0532687216470449i |
In a right-angled triangle, the lengths of the two legs are a = 7 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: 7^2 + 11^2 = c^2
2. Calculate the squares: 49 + 121 = c^2
3. Sum the squares: 170 = c^2
4. Take the square root of both sides: c = √170
Final Ans... |
In a right-angled triangle, the lengths of the two legs are a = 7 and b = 18. 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 + 18^2 = c^2
2. Calculate the squares: 49 + 324 = c^2
3. Sum the squares: 373 = c^2
4. Take the square root of both sides: c = √373
Final Ans... |
Find the roots of the quadratic equation: 7x^2 + -6x + 6 = 0 | 1. Identify coefficients: a=7, b=-6, c=6.
2. Calculate the discriminant: Δ = b^2 - 4ac = -6^2 - 4(7)(6) = 36 - 168 = -132.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (6 ± √132i) / 14.
Final Answer: x = 0.42857142857142855 ± 0.8206518066482898i |
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 |
Find the roots of the quadratic equation: 3x^2 + -5x + -10 = 0 | 1. Identify coefficients: a=3, b=-5, c=-10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 25 - -120 = 145.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (5 ± √145) / 6.
5. Calculate roots: x1 = 2.840265763132049, x2 = -1.1735990964653826.
Final ... |
Compute the indefinite integral of f(x) = 5x^5. | To find the indefinite integral ∫ 5x^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: 5 / 6 = 5/6.
3. Combine and add the constant of integration C.
Final Answer: ∫ 5x^5 dx = 5/6x^6 + C |
Calculate the mean, population variance, and standard deviation for the dataset: [11, 20, 5, 10, 9] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (11 + 20 + 5 + 10 + 9) / 5 = 55 / 5 = 11.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [0.0, 9.0, -6.0, -1.0, -2.0]
Squared Differences: [0.0, 81.0, 36.0, 1.0, 4.0]
Sum... |
Compute the indefinite integral of f(x) = 4x^2. | To find the indefinite integral ∫ 4x^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: 4 / 3 = 4/3.
3. Combine and add the constant of integration C.
Final Answer: ∫ 4x^2 dx = 4/3x^3 + C |
Add the fractions 5/6 and 12/12. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 6 and 12, which is 12.
2. Convert 5/6 by multiplying top and bottom by 2: 10/12.
3. Convert 12/12 by multiplying top and bottom by 1: 12/12.
4. Add the numerators: 10 + 12 = 22.
5. The resulting fraction is 22/12.
6. Simplify by ... |
Compute the quotient and remainder of 67732 divided by 78. Show the long division steps. | To divide 67732 by 78:
1. Determine how many times 78 goes into the leading digits of 67732.
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: 67732 / 78 = 868 with a remainder of 28.
Proof: (868 * 78) ... |
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 +... |
Compute the quotient and remainder of 13874 divided by 18. Show the long division steps. | To divide 13874 by 18:
1. Determine how many times 18 goes into the leading digits of 13874.
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: 13874 / 18 = 770 with a remainder of 14.
Proof: (770 * 18) ... |
In a right-angled triangle, the lengths of the two legs are a = 4 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: 4^2 + 7^2 = c^2
2. Calculate the squares: 16 + 49 = c^2
3. Sum the squares: 65 = c^2
4. Take the square root of both sides: c = √65
Final Answer:... |
In a right-angled triangle, the lengths of the two legs are a = 6 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: 6^2 + 15^2 = c^2
2. Calculate the squares: 36 + 225 = c^2
3. Sum the squares: 261 = c^2
4. Take the square root of both sides: c = √261
Final Ans... |
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