This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Okay Benoni, let's tackle these surds and integrals. 2a) Rationalize the following surds then solve. i) (5)/(sqrt(3)) Step 1: Multiply the numerator and denominator by sqrt(3) to rationalize. (5)/(sqrt(3)) × sqrt(3)sqrt(3) = 5sqrt(3)3 The rationalized expression is 5sqrt(3)3. ii) (3)/(sqrt(3)) Step 1: Multiply the numerator and denominator by sqrt(3). (3)/(sqrt(3)) × sqrt(3)sqrt(3) = 3sqrt(3)3 Step 2: Simplify the expression. 3sqrt(3)3 = sqrt(3) The rationalized expression is sqrt(3). iii) (4)/(sqrt(2)) Step 1: Multiply the numerator and denominator by sqrt(2). (4)/(sqrt(2)) × sqrt(2)sqrt(2) = 4sqrt(2)2 Step 2: Simplify the expression. 4sqrt(2)2 = 2sqrt(2) The rationalized expression is 2sqrt(2). iv) (1)/(sqrt(2)-5) Step 1: Multiply the numerator and denominator by the conjugate of the denominator, which is sqrt(2)+sqrt(5). (1)/(sqrt(2)-5) × sqrt(2)+sqrt(5)sqrt(2)+sqrt(5) Step 2: Expand the numerator and denominator. Remember (a-b)(a+b) = a^2-b^2. sqrt(2)+sqrt(5)(sqrt(2))^2 - (sqrt(5))^2 = sqrt(2)+sqrt(5)2 - 5 sqrt(2)+sqrt(5)-3 = -sqrt(2)+sqrt(5)3 The rationalized expression is -sqrt(2)+sqrt(5)3. 2b) Simplify the following Surds. i) sqrt(32) + 9sqrt(8) Step 1: Simplify each surd by finding perfect square factors. sqrt(32) = sqrt(16 × 2) = 4sqrt(2) sqrt(8) = sqrt(4 × 2) = 2sqrt(2) Step 2: Substitute the simplified surds back into the expression and combine like terms. 4sqrt(2) + 9(2sqrt(2)) = 4sqrt(2) + 18sqrt(2) 4sqrt(2) + 18sqrt(2) = 22sqrt(2) ii) 2sqrt(20) + sqrt(125) - 3sqrt(5) Step 1: Simplify each surd. sqrt(20) = sqrt(4 × 5) = 2sqrt(5) sqrt(125) = sqrt(25 × 5) = 5sqrt(5) Step 2: Substitute the simplified surds and combine like terms. 2(2sqrt(5)) + 5sqrt(5) - 3sqrt(5) 4sqrt(5) + 5sqrt(5) - 3sqrt(5) (4+5-3)sqrt(5) = 6sqrt(5) The simplified expression is 6sqrt(5). iii) sqrt(98) - sqrt(32) - 2sqrt(2) Step 1: Simplify each surd. sqrt(98) = sqrt(49 × 2) = 7sqrt(2) sqrt(32) = sqrt(16 × 2) = 4sqrt(2) Step 2: Substitute the simplified surds and combine like terms. 7sqrt(2) - 4sqrt(2) - 2sqrt(2) (7-4-2)sqrt(2) = 1sqrt(2) = sqrt(2) The simplified expression is sqrt(2). 2c) Integrate all with respect to x the following simple equation. i) 8x^2 Step 1: Apply the power rule for integration, x^n dx = x^n+1n+1 + C. 8x^2 \,dx = 8 x^2+12+1 + C = 8 (x^3)/(3) + C The integral is (8)/(3)x^3 + C. ii) (dx)/(3x+2) Step 1: Use the integration rule (1)/(ax+b) dx = (1)/(a) |ax+b| + C. Here a=3 and b=2. (1)/(3x+2) \,dx = (1)/(3) |3x+2| + C The integral is (1)/(3)|3x+2| + C. iii) (x^4+x+4)/(x^3) Step 1: Rewrite the expression by dividing each term in the numerator by x^3. (x^4)/(x^3) + (x)/(x^3) + (4)/(x^3) = x + x^-2 + 4x^-3 Step 2: Integrate each term using the power rule. (x + x^-2 + 4x^-3) \,dx = x\,dx + x^-2\,dx + 4x^-3\,dx = x^1+11+1 + x^-2+1-2+1 + 4x^-3+1-3+1 + C = (x^2)/(2) + x^-1-1 + 4x^-2-2 + C = (x^2)/(2) - (1)/(x) - (4)/(2x^2) + C = (x^2)/(2) - (1)/(x) - (2)/(x^2) + C The integral is (x^2)/(2) - (1)/(x) - (2)/(x^2) + C. 2d) Write down The Conjugate of the Surds. i) sqrt(3)+sqrt(2) The conjugate of sqrt(3)+sqrt(2) is sqrt(3)-sqrt(2). ii) 13+5sqrt(2)7 The conjugate of the surd part 13+5sqrt(2) is 13-5sqrt(2). (The denominator does not change for the conjugate of the expression itself, only the sign of the surd term in the numerator). iii) 1-2+sqrt(5) Step 1: Simplify the expression first. 1-2+sqrt(5) = -1+sqrt(5) Step 2: Write down the conjugate. The conjugate of -1+sqrt(5) is -1-sqrt(5). iv) 5sqrt(7)+5sqrt(3)+sqrt(7)-sqrt(3) Step 1: Simplify the expression by combining like terms. (5sqrt(7)+sqrt(7)) + (5sqrt(3)-sqrt(3)) = 6sqrt(7) + 4sqrt(3) Step 2: Write down the conjugate. The conjugate of 6sqrt(7)+4sqrt(3) is 6sqrt(7)-4sqrt(3). That's 2 down. 3 left today — send the next one.