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Compute the quotient and remainder of 53208 divided by 61. Show the long division steps. | To divide 53208 by 61:
1. Determine how many times 61 goes into the leading digits of 53208.
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: 53208 / 61 = 872 with a remainder of 16.
Proof: (872 * 61) ... |
Evaluate sin(0°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(0°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 0.0000 (exact form depends on root simplification if applicable) |
Compute the quotient and remainder of 12240 divided by 48. Show the long division steps. | To divide 12240 by 48:
1. Determine how many times 48 goes into the leading digits of 12240.
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: 12240 / 48 = 255 with a remainder of 0.
Proof: (255 * 48) +... |
In a right-angled triangle, the lengths of the two legs are a = 17 and b = 5. Find the length of the hypotenuse c. | According to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides:
a^2 + b^2 = c^2
1. Substitute the known values: 17^2 + 5^2 = c^2
2. Calculate the squares: 289 + 25 = c^2
3. Sum the squares: 314 = c^2
4. Take the square root of both sides: c = √314
Final Ans... |
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... |
In a right-angled triangle, the lengths of the two legs are a = 4 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: 4^2 + 17^2 = c^2
2. Calculate the squares: 16 + 289 = c^2
3. Sum the squares: 305 = c^2
4. Take the square root of both sides: c = √305
Final Ans... |
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 roots of the quadratic equation: 10x^2 + -9x + -1 = 0 | 1. Identify coefficients: a=10, b=-9, c=-1.
2. Calculate the discriminant: Δ = b^2 - 4ac = 81 - -40 = 121.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (9 ± √121) / 20.
5. Calculate roots: x1 = 1.0, x2 = -0.1.
Final Answer: x = 1.0, -0.1 |
Find the roots of the quadratic equation: 6x^2 + -2x + -9 = 0 | 1. Identify coefficients: a=6, b=-2, c=-9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 4 - -216 = 220.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (2 ± √220) / 12.
5. Calculate roots: x1 = 1.402699747849277, x2 = -1.0693664145159438.
Final A... |
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) |
Compute the quotient and remainder of 51279 divided by 11. Show the long division steps. | To divide 51279 by 11:
1. Determine how many times 11 goes into the leading digits of 51279.
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: 51279 / 11 = 4661 with a remainder of 8.
Proof: (4661 * 11)... |
Calculate the mean, population variance, and standard deviation for the dataset: [16, 4, 4, 10, 14] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (16 + 4 + 4 + 10 + 14) / 5 = 48 / 5 = 9.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [6.4, -5.6, -5.6, 0.4, 4.4]
Squared Differences: [40.96, 31.36, 31.36, 0.16, 19.36]
... |
Find the roots of the quadratic equation: 6x^2 + -2x + -2 = 0 | 1. Identify coefficients: a=6, b=-2, c=-2.
2. Calculate the discriminant: Δ = b^2 - 4ac = 4 - -48 = 52.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (2 ± √52) / 12.
5. Calculate roots: x1 = 0.7675918792439983, x2 = -0.4342585459106649.
Final Ans... |
Add the fractions 10/9 and 5/7. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 9 and 7, which is 63.
2. Convert 10/9 by multiplying top and bottom by 7: 70/63.
3. Convert 5/7 by multiplying top and bottom by 9: 45/63.
4. Add the numerators: 70 + 45 = 115.
5. The resulting fraction is 115/63.
Final Answer: 1... |
Multiply the following 2x2 matrices:
A =
[-2, 2]
[0, 3]
B =
[4, -5]
[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) = (-2*4) + (2*2) = -4
C12 = (A11 * B12) + (A12 * B22) = (-2*-5) + (2*-3) = 4
C21 = (A21 * B11) + (A22 * B21) = (0*4) + (3*2) = 6
C22 = (A21 * B12) + (A22 * B22) = (0*-5) + (3*-3) = -9
Result Ma... |
In a right-angled triangle, the lengths of the two legs are a = 14 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: 14^2 + 4^2 = c^2
2. Calculate the squares: 196 + 16 = c^2
3. Sum the squares: 212 = c^2
4. Take the square root of both sides: c = √212
Final Ans... |
Multiply the following 2x2 matrices:
A =
[0, 3]
[5, -5]
B =
[-3, 3]
[0, 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*-3) + (3*0) = 0
C12 = (A11 * B12) + (A12 * B22) = (0*3) + (3*2) = 6
C21 = (A21 * B11) + (A22 * B21) = (5*-3) + (-5*0) = -15
C22 = (A21 * B12) + (A22 * B22) = (5*3) + (-5*2) = 5
Result Matr... |
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 |
Compute the quotient and remainder of 59203 divided by 84. Show the long division steps. | To divide 59203 by 84:
1. Determine how many times 84 goes into the leading digits of 59203.
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: 59203 / 84 = 704 with a remainder of 67.
Proof: (704 * 84) ... |
Multiply the following 2x2 matrices:
A =
[-3, 5]
[5, -3]
B =
[0, -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) = (-3*0) + (5*4) = 20
C12 = (A11 * B12) + (A12 * B22) = (-3*-1) + (5*0) = 3
C21 = (A21 * B11) + (A22 * B21) = (5*0) + (-3*4) = -12
C22 = (A21 * B12) + (A22 * B22) = (5*-1) + (-3*0) = -5
Result ... |
Calculate the mean, population variance, and standard deviation for the dataset: [4, 4, 14, 4, 7] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (4 + 4 + 14 + 4 + 7) / 5 = 33 / 5 = 6.6
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-2.6, -2.6, 7.4, -2.6, 0.4]
Squared Differences: [6.76, 6.76, 54.76, 6.76, 0.16]
Su... |
Add the fractions 14/10 and 1/2. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 10 and 2, which is 10.
2. Convert 14/10 by multiplying top and bottom by 1: 14/10.
3. Convert 1/2 by multiplying top and bottom by 5: 5/10.
4. Add the numerators: 14 + 5 = 19.
5. The resulting fraction is 19/10.
Final Answer: 19/... |
Add the fractions 7/14 and 3/3. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 14 and 3, which is 42.
2. Convert 7/14 by multiplying top and bottom by 3: 21/42.
3. Convert 3/3 by multiplying top and bottom by 14: 42/42.
4. Add the numerators: 21 + 42 = 63.
5. The resulting fraction is 63/42.
6. Simplify by ... |
Add the fractions 8/12 and 3/14. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 12 and 14, which is 84.
2. Convert 8/12 by multiplying top and bottom by 7: 56/84.
3. Convert 3/14 by multiplying top and bottom by 6: 18/84.
4. Add the numerators: 56 + 18 = 74.
5. The resulting fraction is 74/84.
6. Simplify by... |
Calculate the mean, population variance, and standard deviation for the dataset: [18, 15, 3, 8, 18] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (18 + 15 + 3 + 8 + 18) / 5 = 62 / 5 = 12.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [5.6, 2.6, -9.4, -4.4, 5.6]
Squared Differences: [31.36, 6.76, 88.36, 19.36, 31.36]
... |
Add the fractions 4/9 and 11/4. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 9 and 4, which is 36.
2. Convert 4/9 by multiplying top and bottom by 4: 16/36.
3. Convert 11/4 by multiplying top and bottom by 9: 99/36.
4. Add the numerators: 16 + 99 = 115.
5. The resulting fraction is 115/36.
Final Answer: 1... |
Calculate the mean, population variance, and standard deviation for the dataset: [11, 10, 2, 4, 10] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (11 + 10 + 2 + 4 + 10) / 5 = 37 / 5 = 7.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [3.6, 2.6, -5.4, -3.4, 2.6]
Squared Differences: [12.96, 6.76, 29.16, 11.56, 6.76]
... |
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 ... |
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 ... |
Compute the indefinite integral of f(x) = 6x^3. | To find the indefinite integral ∫ 6x^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: 6 / 4 = 3/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 6x^3 dx = 3/2x^4 + C |
In a right-angled triangle, the lengths of the two legs are a = 8 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: 8^2 + 19^2 = c^2
2. Calculate the squares: 64 + 361 = c^2
3. Sum the squares: 425 = c^2
4. Take the square root of both sides: c = √425
Final Ans... |
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) |
Compute the quotient and remainder of 11138 divided by 76. Show the long division steps. | To divide 11138 by 76:
1. Determine how many times 76 goes into the leading digits of 11138.
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: 11138 / 76 = 146 with a remainder of 42.
Proof: (146 * 76) ... |
Compute the indefinite integral of f(x) = 7x^3. | To find the indefinite integral ∫ 7x^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: 7 / 4 = 7/4.
3. Combine and add the constant of integration C.
Final Answer: ∫ 7x^3 dx = 7/4x^4 + C |
Add the fractions 3/15 and 15/11. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 15 and 11, which is 165.
2. Convert 3/15 by multiplying top and bottom by 11: 33/165.
3. Convert 15/11 by multiplying top and bottom by 15: 225/165.
4. Add the numerators: 33 + 225 = 258.
5. The resulting fraction is 258/165.
6. ... |
Find the roots of the quadratic equation: 6x^2 + 0x + 6 = 0 | 1. Identify coefficients: a=6, b=0, c=6.
2. Calculate the discriminant: Δ = b^2 - 4ac = 0^2 - 4(6)(6) = 0 - 144 = -144.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (0 ± √144i) / 12.
Final Answer: x = 0.0 ± 1.0i |
Find the roots of the quadratic equation: 10x^2 + -7x + 2 = 0 | 1. Identify coefficients: a=10, b=-7, c=2.
2. Calculate the discriminant: Δ = b^2 - 4ac = -7^2 - 4(10)(2) = 49 - 80 = -31.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (7 ± √31i) / 20.
Final Answer: x = 0.35 ± 0.27838821814150105i |
Add the fractions 7/7 and 10/9. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 7 and 9, which is 63.
2. Convert 7/7 by multiplying top and bottom by 9: 63/63.
3. Convert 10/9 by multiplying top and bottom by 7: 70/63.
4. Add the numerators: 63 + 70 = 133.
5. The resulting fraction is 133/63.
6. Simplify by ... |
Compute the indefinite integral of f(x) = 2x^3. | To find the indefinite integral ∫ 2x^3 dx, we use the power rule for integration:
∫ x^n dx = x^(n+1) / (n+1) + C (where n ≠ -1).
1. Add 1 to the exponent: 3 + 1 = 4.
2. Divide the coefficient by the new exponent: 2 / 4 = 1/2.
3. Combine and add the constant of integration C.
Final Answer: ∫ 2x^3 dx = 1/2x^4 + C |
Calculate the mean, population variance, and standard deviation for the dataset: [14, 11, 6, 18, 18] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (14 + 11 + 6 + 18 + 18) / 5 = 67 / 5 = 13.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [0.6, -2.4, -7.4, 4.6, 4.6]
Squared Differences: [0.36, 5.76, 54.76, 21.16, 21.16]
... |
Multiply the following 2x2 matrices:
A =
[2, -2]
[4, 5]
B =
[-3, -5]
[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) = (2*-3) + (-2*3) = -12
C12 = (A11 * B12) + (A12 * B22) = (2*-5) + (-2*4) = -18
C21 = (A21 * B11) + (A22 * B21) = (4*-3) + (5*3) = 3
C22 = (A21 * B12) + (A22 * B22) = (4*-5) + (5*4) = 0
Result ... |
Evaluate the logarithm: log_4(1024) | To evaluate log_4(1024), we ask the question: '4 raised to what power equals 1024?'
Let x be the unknown power: 4^x = 1024
Since 4^5 = 1024, it follows that x = 5.
Final Answer: 5 |
Evaluate sin(90°) without a calculator. Provide the exact value. | Using standard trigonometric values from the unit circle:
The exact value of sin(90°) is derived from the properties of special right triangles (30-60-90 or 45-45-90).
Final Answer: 1.0000 (exact form depends on root simplification if applicable) |
Compute the indefinite integral of f(x) = 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 |
In a right-angled triangle, the lengths of the two legs are a = 15 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: 15^2 + 11^2 = c^2
2. Calculate the squares: 225 + 121 = c^2
3. Sum the squares: 346 = c^2
4. Take the square root of both sides: c = √346
Final A... |
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: 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 |
Compute the quotient and remainder of 51250 divided by 8. Show the long division steps. | To divide 51250 by 8:
1. Determine how many times 8 goes into the leading digits of 51250.
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: 51250 / 8 = 6406 with a remainder of 2.
Proof: (6406 * 8) + 2... |
Find the roots of the quadratic equation: 4x^2 + 8x + 10 = 0 | 1. Identify coefficients: a=4, b=8, c=10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 8^2 - 4(4)(10) = 64 - 160 = -96.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (-8 ± √96i) / 8.
Final Answer: x = -1.0 ± 1.224744871391589i |
Multiply the following 2x2 matrices:
A =
[2, -2]
[-4, 3]
B =
[-1, -5]
[-5, 3] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (2*-1) + (-2*-5) = 8
C12 = (A11 * B12) + (A12 * B22) = (2*-5) + (-2*3) = -16
C21 = (A21 * B11) + (A22 * B21) = (-4*-1) + (3*-5) = -11
C22 = (A21 * B12) + (A22 * B22) = (-4*-5) + (3*3) = 29
Re... |
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_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 |
Find the roots of the quadratic equation: 10x^2 + -5x + -10 = 0 | 1. Identify coefficients: a=10, b=-5, c=-10.
2. Calculate the discriminant: Δ = b^2 - 4ac = 25 - -400 = 425.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (5 ± √425) / 20.
5. Calculate roots: x1 = 1.2807764064044151, x2 = -0.7807764064044151.
Fin... |
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 = 14 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: 14^2 + 16^2 = c^2
2. Calculate the squares: 196 + 256 = c^2
3. Sum the squares: 452 = c^2
4. Take the square root of both sides: c = √452
Final A... |
In a right-angled triangle, the lengths of the two legs are a = 20 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: 20^2 + 13^2 = c^2
2. Calculate the squares: 400 + 169 = c^2
3. Sum the squares: 569 = c^2
4. Take the square root of both sides: c = √569
Final A... |
Multiply the following 2x2 matrices:
A =
[-4, -5]
[3, -1]
B =
[1, -1]
[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) = (-4*1) + (-5*1) = -9
C12 = (A11 * B12) + (A12 * B22) = (-4*-1) + (-5*-1) = 9
C21 = (A21 * B11) + (A22 * B21) = (3*1) + (-1*1) = 2
C22 = (A21 * B12) + (A22 * B22) = (3*-1) + (-1*-1) = -2
Resul... |
In a right-angled triangle, the lengths of the two legs are a = 15 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: 15^2 + 15^2 = c^2
2. Calculate the squares: 225 + 225 = c^2
3. Sum the squares: 450 = c^2
4. Take the square root of both sides: c = √450
Final A... |
In a right-angled triangle, the lengths of the two legs are a = 5 and b = 9. 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 + 9^2 = c^2
2. Calculate the squares: 25 + 81 = c^2
3. Sum the squares: 106 = c^2
4. Take the square root of both sides: c = √106
Final Answe... |
In a right-angled triangle, the lengths of the two legs are a = 16 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: 16^2 + 11^2 = c^2
2. Calculate the squares: 256 + 121 = c^2
3. Sum the squares: 377 = c^2
4. Take the square root of both sides: c = √377
Final A... |
Compute the quotient and remainder of 68190 divided by 6. Show the long division steps. | To divide 68190 by 6:
1. Determine how many times 6 goes into the leading digits of 68190.
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: 68190 / 6 = 11365 with a remainder of 0.
Proof: (11365 * 6) +... |
Add the fractions 15/3 and 13/11. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 3 and 11, which is 33.
2. Convert 15/3 by multiplying top and bottom by 11: 165/33.
3. Convert 13/11 by multiplying top and bottom by 3: 39/33.
4. Add the numerators: 165 + 39 = 204.
5. The resulting fraction is 204/33.
6. Simpli... |
Evaluate the logarithm: log_2(16) | To evaluate log_2(16), we ask the question: '2 raised to what power equals 16?'
Let x be the unknown power: 2^x = 16
Since 2^4 = 16, it follows that x = 4.
Final Answer: 4 |
Add the fractions 1/2 and 7/2. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 2 and 2, which is 2.
2. Convert 1/2 by multiplying top and bottom by 1: 1/2.
3. Convert 7/2 by multiplying top and bottom by 1: 7/2.
4. Add the numerators: 1 + 7 = 8.
5. The resulting fraction is 8/2.
6. Simplify by dividing nume... |
Compute the indefinite integral of f(x) = 7x^3. | To find the indefinite integral ∫ 7x^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: 7 / 4 = 7/4.
3. Combine and add the constant of integration C.
Final Answer: ∫ 7x^3 dx = 7/4x^4 + C |
Compute the quotient and remainder of 22531 divided by 67. Show the long division steps. | To divide 22531 by 67:
1. Determine how many times 67 goes into the leading digits of 22531.
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: 22531 / 67 = 336 with a remainder of 19.
Proof: (336 * 67) ... |
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 ... |
Evaluate the logarithm: log_4(256) | To evaluate log_4(256), we ask the question: '4 raised to what power equals 256?'
Let x be the unknown power: 4^x = 256
Since 4^4 = 256, it follows that x = 4.
Final Answer: 4 |
Find the roots of the quadratic equation: 3x^2 + -2x + 8 = 0 | 1. Identify coefficients: a=3, b=-2, c=8.
2. Calculate the discriminant: Δ = b^2 - 4ac = -2^2 - 4(3)(8) = 4 - 96 = -92.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (2 ± √92i) / 6.
Final Answer: x = 0.3333333333333333 ± 1.5986105077709063i |
Compute the quotient and remainder of 35936 divided by 61. Show the long division steps. | To divide 35936 by 61:
1. Determine how many times 61 goes into the leading digits of 35936.
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: 35936 / 61 = 589 with a remainder of 7.
Proof: (589 * 61) +... |
Compute the quotient and remainder of 2113 divided by 75. Show the long division steps. | To divide 2113 by 75:
1. Determine how many times 75 goes into the leading digits of 2113.
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: 2113 / 75 = 28 with a remainder of 13.
Proof: (28 * 75) + 13 ... |
Find the roots of the quadratic equation: 8x^2 + -4x + -9 = 0 | 1. Identify coefficients: a=8, b=-4, c=-9.
2. Calculate the discriminant: Δ = b^2 - 4ac = 16 - -288 = 304.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (4 ± √304) / 16.
5. Calculate roots: x1 = 1.3397247358851685, x2 = -0.8397247358851685.
Final... |
In a right-angled triangle, the lengths of the two legs are a = 16 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: 16^2 + 17^2 = c^2
2. Calculate the squares: 256 + 289 = c^2
3. Sum the squares: 545 = c^2
4. Take the square root of both sides: c = √545
Final A... |
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 |
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) |
In a right-angled triangle, the lengths of the two legs are a = 16 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: 16^2 + 19^2 = c^2
2. Calculate the squares: 256 + 361 = c^2
3. Sum the squares: 617 = c^2
4. Take the square root of both sides: c = √617
Final A... |
Multiply the following 2x2 matrices:
A =
[4, 3]
[3, -4]
B =
[-4, 2]
[-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) = (4*-4) + (3*-1) = -19
C12 = (A11 * B12) + (A12 * B22) = (4*2) + (3*-1) = 5
C21 = (A21 * B11) + (A22 * B21) = (3*-4) + (-4*-1) = -8
C22 = (A21 * B12) + (A22 * B22) = (3*2) + (-4*-1) = 10
Resul... |
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: [20, 20, 3, 5, 1] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (20 + 20 + 3 + 5 + 1) / 5 = 49 / 5 = 9.8
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [10.2, 10.2, -6.8, -4.8, -8.8]
Squared Differences: [104.04, 104.04, 46.24, 23.04, 77.... |
Add the fractions 2/7 and 5/6. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 7 and 6, which is 42.
2. Convert 2/7 by multiplying top and bottom by 6: 12/42.
3. Convert 5/6 by multiplying top and bottom by 7: 35/42.
4. Add the numerators: 12 + 35 = 47.
5. The resulting fraction is 47/42.
Final Answer: 47/4... |
Multiply the following 2x2 matrices:
A =
[-5, 0]
[-5, 2]
B =
[2, 4]
[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) = (-5*2) + (0*1) = -10
C12 = (A11 * B12) + (A12 * B22) = (-5*4) + (0*4) = -20
C21 = (A21 * B11) + (A22 * B21) = (-5*2) + (2*1) = -8
C22 = (A21 * B12) + (A22 * B22) = (-5*4) + (2*4) = -12
Result... |
Multiply the following 2x2 matrices:
A =
[5, -4]
[-5, 0]
B =
[1, 2]
[-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) = (5*1) + (-4*-2) = 13
C12 = (A11 * B12) + (A12 * B22) = (5*2) + (-4*-3) = 22
C21 = (A21 * B11) + (A22 * B21) = (-5*1) + (0*-2) = -5
C22 = (A21 * B12) + (A22 * B22) = (-5*2) + (0*-3) = -10
Resu... |
Find the roots of the quadratic equation: 6x^2 + 3x + -8 = 0 | 1. Identify coefficients: a=6, b=3, c=-8.
2. Calculate the discriminant: Δ = b^2 - 4ac = 9 - -192 = 201.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (-3 ± √201) / 12.
5. Calculate roots: x1 = 0.9314539065631521, x2 = -1.4314539065631522.
Final ... |
Add the fractions 9/13 and 6/15. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 13 and 15, which is 195.
2. Convert 9/13 by multiplying top and bottom by 15: 135/195.
3. Convert 6/15 by multiplying top and bottom by 13: 78/195.
4. Add the numerators: 135 + 78 = 213.
5. The resulting fraction is 213/195.
6. S... |
Evaluate the logarithm: log_4(1024) | To evaluate log_4(1024), we ask the question: '4 raised to what power equals 1024?'
Let x be the unknown power: 4^x = 1024
Since 4^5 = 1024, it follows that x = 5.
Final Answer: 5 |
Calculate the mean, population variance, and standard deviation for the dataset: [3, 7, 1, 9, 10] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (3 + 7 + 1 + 9 + 10) / 5 = 30 / 5 = 6.0
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-3.0, 1.0, -5.0, 3.0, 4.0]
Squared Differences: [9.0, 1.0, 25.0, 9.0, 16.0]
Sum of ... |
Find the roots of the quadratic equation: 7x^2 + -3x + 2 = 0 | 1. Identify coefficients: a=7, b=-3, c=2.
2. Calculate the discriminant: Δ = b^2 - 4ac = -3^2 - 4(7)(2) = 9 - 56 = -47.
3. Since Δ < 0, the equation has two complex (imaginary) roots.
4. Roots are given by x = (-b ± √Δ) / 2a = (3 ± √47i) / 14.
Final Answer: x = 0.21428571428571427 ± 0.48968961431436026i |
Multiply the following 2x2 matrices:
A =
[-3, 5]
[4, -1]
B =
[2, -5]
[-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*2) + (5*-5) = -31
C12 = (A11 * B12) + (A12 * B22) = (-3*-5) + (5*-1) = 10
C21 = (A21 * B11) + (A22 * B21) = (4*2) + (-1*-5) = 13
C22 = (A21 * B12) + (A22 * B22) = (4*-5) + (-1*-1) = -19
R... |
Multiply the following 2x2 matrices:
A =
[-4, 3]
[4, 2]
B =
[1, 3]
[2, -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*1) + (3*2) = 2
C12 = (A11 * B12) + (A12 * B22) = (-4*3) + (3*-2) = -18
C21 = (A21 * B11) + (A22 * B21) = (4*1) + (2*2) = 8
C22 = (A21 * B12) + (A22 * B22) = (4*3) + (2*-2) = 8
Result Matr... |
Add the fractions 12/8 and 5/13. Simplify the result. | To add fractions, we need a common denominator.
1. Find the Least Common Multiple (LCM) of 8 and 13, which is 104.
2. Convert 12/8 by multiplying top and bottom by 13: 156/104.
3. Convert 5/13 by multiplying top and bottom by 8: 40/104.
4. Add the numerators: 156 + 40 = 196.
5. The resulting fraction is 196/104.
6. Sim... |
Multiply the following 2x2 matrices:
A =
[1, 4]
[-2, -3]
B =
[1, -5]
[3, 3] | To multiply matrices A and B, we take the dot product of the rows of A with the columns of B.
C11 = (A11 * B11) + (A12 * B21) = (1*1) + (4*3) = 13
C12 = (A11 * B12) + (A12 * B22) = (1*-5) + (4*3) = 7
C21 = (A21 * B11) + (A22 * B21) = (-2*1) + (-3*3) = -11
C22 = (A21 * B12) + (A22 * B22) = (-2*-5) + (-3*3) = 1
Result M... |
In a right-angled triangle, the lengths of the two legs are a = 7 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: 7^2 + 15^2 = c^2
2. Calculate the squares: 49 + 225 = c^2
3. Sum the squares: 274 = c^2
4. Take the square root of both sides: c = √274
Final Ans... |
Evaluate the logarithm: log_3(243) | To evaluate log_3(243), we ask the question: '3 raised to what power equals 243?'
Let x be the unknown power: 3^x = 243
Since 3^5 = 243, it follows that x = 5.
Final Answer: 5 |
Evaluate the logarithm: log_4(16) | To evaluate log_4(16), we ask the question: '4 raised to what power equals 16?'
Let x be the unknown power: 4^x = 16
Since 4^2 = 16, it follows that x = 2.
Final Answer: 2 |
In a right-angled triangle, the lengths of the two legs are a = 20 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: 20^2 + 13^2 = c^2
2. Calculate the squares: 400 + 169 = c^2
3. Sum the squares: 569 = c^2
4. Take the square root of both sides: c = √569
Final A... |
Compute the quotient and remainder of 21685 divided by 54. Show the long division steps. | To divide 21685 by 54:
1. Determine how many times 54 goes into the leading digits of 21685.
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: 21685 / 54 = 401 with a remainder of 31.
Proof: (401 * 54) ... |
Find the roots of the quadratic equation: 7x^2 + -3x + -5 = 0 | 1. Identify coefficients: a=7, b=-3, c=-5.
2. Calculate the discriminant: Δ = b^2 - 4ac = 9 - -140 = 149.
3. Since Δ ≥ 0, the equation has real roots. Apply the quadratic formula: x = (-b ± √Δ) / 2a.
4. Substitute values: x = (3 ± √149) / 14.
5. Calculate roots: x1 = 1.0861825439809787, x2 = -0.6576111154095502.
Final ... |
Compute the quotient and remainder of 97674 divided by 85. Show the long division steps. | To divide 97674 by 85:
1. Determine how many times 85 goes into the leading digits of 97674.
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: 97674 / 85 = 1149 with a remainder of 9.
Proof: (1149 * 85)... |
Compute the quotient and remainder of 60717 divided by 3. Show the long division steps. | To divide 60717 by 3:
1. Determine how many times 3 goes into the leading digits of 60717.
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: 60717 / 3 = 20239 with a remainder of 0.
Proof: (20239 * 3) +... |
Calculate the mean, population variance, and standard deviation for the dataset: [6, 19, 15, 2, 20] | 1. Calculate the mean (μ): Sum all values and divide by N (5).
μ = (6 + 19 + 15 + 2 + 20) / 5 = 62 / 5 = 12.4
2. Calculate the population variance (σ^2): Find the average of the squared differences from the Mean.
Differences: [-6.4, 6.6, 2.6, -10.4, 7.6]
Squared Differences: [40.96, 43.56, 6.76, 108.16, 57.76... |
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