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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Here are the answers to your questions: a) Simplify sqrt(18) + sqrt(50) - sqrt(8). Step 1: Simplify each surd individually by finding the largest perfect square factor. sqrt(18) = sqrt(9 × 2) = sqrt(9) × sqrt(2) = 3sqrt(2) sqrt(50) = sqrt(25 × 2) = sqrt(25) × sqrt(2) = 5sqrt(2) sqrt(8) = sqrt(4 × 2) = sqrt(4) × sqrt(2) = 2sqrt(2) Step 2: Substitute the simplified surds back into the expression. 3sqrt(2) + 5sqrt(2) - 2sqrt(2) Step 3: Combine the like surds. (3 + 5 - 2)sqrt(2) (8 - 2)sqrt(2) 6sqrt(2) b) Given that sqrt(2) = 1.414 and sqrt(3) = 1.732, evaluate (1)/(sqrt(3) - 2) without using tables or a calculator. Step 1: Rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator. The conjugate of sqrt(3) - sqrt(2) is sqrt(3) + sqrt(2). (1)/(sqrt(3) - 2) × sqrt(3) + sqrt(2)sqrt(3) + sqrt(2) Step 2: Apply the difference of squares formula (a-b)(a+b) = a^2 - b^2 to the denominator. sqrt(3) + sqrt(2)(sqrt(3))^2 - (sqrt(2))^2 sqrt(3) + sqrt(2)3 - 2 sqrt(3) + sqrt(2)1 = sqrt(3) + sqrt(2) Step 3: Substitute the given approximate values for sqrt(3) and sqrt(2). 1.732 + 1.414 Step 4: Perform the addition. 3.146 c) Solve for x: 3^2x-1 = 27. Step 1: Express both sides of the equation with the same base. 3^2x-1 = 3^3 Step 2: Equate the exponents since the bases are the same. 2x - 1 = 3 Step 3: Solve the linear equation for x. 2x = 3 + 1 2x = 4 x = (4)/(2) x = 2 --- Notes on Topics Involved in Part b: 1. Rationalizing the Denominator This is a key technique used to remove surds from the denominator of a fraction, making the expression simpler and easier to work with, especially for numerical evaluation. Case 1: Denominator is a single surd term (e.g., sqrt(a)) Multiply both the numerator and the denominator by the surd in the denominator. Example: Rationalize (7)/(sqrt(3)). (7)/(sqrt(3)) × sqrt(3)sqrt(3) = 7sqrt(3)3 Case 2: Denominator is a binomial surd (e.g., a + sqrt(b) or sqrt(a) + sqrt(b)) Multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate is formed by changing the sign between the terms. This method uses the difference of squares identity: (x+y)(x-y) = x^2 - y^2, which eliminates the surds from the denominator. • The conjugate of a + sqrt(b) is a - sqrt(b). • The conjugate of sqrt(a) - sqrt(b) is sqrt(a) + sqrt(b). Example: Rationalize (5)/(2 + sqrt(7)). Step 1: Identify the conjugate of the denominator. The conjugate of 2 + sqrt(7) is 2 - sqrt(7). Step 2: Multiply the numerator and denominator by the conjugate. (5)/(2 + sqrt(7)) × 2 - sqrt(7)2 - sqrt(7) Step 3: Perform the multiplication. 5(2 - sqrt(7))2^2 - (sqrt(7))^2 10 - 5sqrt(7)4 - 7 10 - 5sqrt(7)-3 = -10 - 5sqrt(7)3 or 5sqrt(7) - 103 2. Substitution of Approximate Values When asked to evaluate an expression involving surds using given approximate values (e.g., sqrt(2) ≈ 1.414), it is crucial to: • Simplify the expression completely first, including rationalizing any denominators. • Then, substitute the approximate values into the simplified expression. This approach minimizes calculation errors and ensures accuracy, especially when dealing with complex fractions. Example: Evaluate (1)/(sqrt(6) - 5) given sqrt(6) ≈ 2.449 and sqrt(5) ≈ 2.236. Step 1: Rationalize the denominator. The conjugate of sqrt(6) - sqrt(5) is sqrt(6) + sqrt(5). (1)/(sqrt(6) - 5) × sqrt(6) + sqrt(5)sqrt(6) + sqrt(5) sqrt(6) + sqrt(5)(sqrt(6))^2 - (sqrt(5))^2 sqrt(6) + sqrt(5)6 - 5 sqrt(6) + sqrt(5)1 = sqrt(6) + sqrt(5) Step 2: Substitute the given approximate values. 2.449 + 2.236 = 4.685 Last free one today — make it count tomorrow, or type /upgrade for unlimited.