1. Let's continue with surds, which are expressions containing roots, such as square roots, cube roots, etc. 2. The main rules for surds are: - $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$ - $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$ - $\sqrt{a^2} = |a|$ 3. Simplifying surds involves factoring the number inside the root into perfect squares and other factors. 4. For example, simplify $\sqrt{50}$: - Factor 50 as $25 \times 2$ - Use the rule $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$ to write $\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2}$ - Since $\sqrt{25} = 5$, the expression simplifies to $5\sqrt{2}$ 5. Another example: simplify $\frac{\sqrt{18}}{\sqrt{2}}$: - Use the rule $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$ to get $\sqrt{\frac{18}{2}} = \sqrt{9}$ - Since $\sqrt{9} = 3$, the expression simplifies to 3 6. Rationalizing the denominator means removing surds from the denominator by multiplying numerator and denominator by a suitable surd. 7. For example, rationalize $\frac{3}{\sqrt{5}}$: - Multiply numerator and denominator by $\sqrt{5}$ to get $\frac{3 \times \sqrt{5}}{\sqrt{5} \times \sqrt{5}} = \frac{3\sqrt{5}}{5}$ This concludes the continuation with surds and their simplification.