Section A: Rules Surd Rules 1. Multiplication Rule: \( \sqrt{a} \times \sqrt{b}=\sqrt{a \times b} \) 2. Division Rule: \( \frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}} \) 3. Addition and Subtraction: Surds can only be added or subtracted if they have the same radicand: \[ 3 \sqrt{2}+4 \sqrt{2}=7 \sqrt{2} \] 4. Simplifying Surds: \( \sqrt{a b}=\sqrt{a} \times \sqrt{b} \) always written in the simplest form where possible. 5. Rationalizing the Denominator: Multiply the numerator and denominator by a suitable surd to remove the square root from thedenominator. The following surds rules will help in simplifying surds and other mathematical expressions: Simplifving Surds: Example: Simplify \( \sqrt{200} \) Steps: 1. Factorize 200 into a perfect square and another factor(Prime number): \[ \sqrt{200}=\sqrt{100 \times 2} \] 2. Split the square root: \[ \sqrt{100} \times \sqrt{2}=10 \sqrt{2} \] 3. Final answer: \[ 10 \sqrt{2} \]

Let's apply the surd rules to simplify the expression \( \sqrt{200} \) step by step, as outlined in the example provided. ### Step 1: Factorization We start by factorizing \( 200 \) into a perfect square and another factor. The perfect square that divides \( 200 \) is \( 100 \), which is \( 10^2 \). Thus, we can express \( 200 \) as: \[ 200 = 100 \times 2 \] ### Step 2: Splitting the Square Root Using the multiplication rule for surds, we can split the square root: \[ \sqrt{200} = \sqrt{100 \times 2} = \sqrt{100} \times \sqrt{2} \] ### Step 3: Simplifying the Square Root Now, we simplify \( \sqrt{100} \): \[ \sqrt{100} = 10 \] So, substituting this back into our expression gives: \[ \sqrt{200} = 10 \times \sqrt{2} \] ### Final Answer Thus, the simplified form of \( \sqrt{200} \) is: \[ 10 \sqrt{2} \] This process demonstrates the application of the surd rules effectively, leading us to the final simplified expression.