1. The problem is to understand the topic of Surds, which are irrational roots that cannot be simplified to remove the root. 2. A surd is an expression containing a root, such as $\sqrt{2}$ or $\sqrt{3}$, that cannot be simplified to a rational number. 3. Important rules: - $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$ - $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$ - You cannot add or subtract surds unless they have the same root part (like terms). 4. Example: Simplify $\sqrt{50}$. - Factor 50 as $25 \times 2$. - Use $\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2}$. 5. Example: Simplify $3\sqrt{8} + 2\sqrt{18}$. - Simplify each surd: $3\sqrt{8} = 3 \times \sqrt{4 \times 2} = 3 \times 2\sqrt{2} = 6\sqrt{2}$ $2\sqrt{18} = 2 \times \sqrt{9 \times 2} = 2 \times 3\sqrt{2} = 6\sqrt{2}$ - Add like surds: $6\sqrt{2} + 6\sqrt{2} = 12\sqrt{2}$. 6. Example: Rationalize the denominator of $\frac{5}{\sqrt{3}}$. - Multiply numerator and denominator by $\sqrt{3}$: $\frac{5}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{5\sqrt{3}}{3}$. 7. Surds are useful in exact answers where decimals are not precise. Final answer examples: - $\sqrt{50} = 5\sqrt{2}$ - $3\sqrt{8} + 2\sqrt{18} = 12\sqrt{2}$ - $\frac{5}{\sqrt{3}} = \frac{5\sqrt{3}}{3}$ This explanation covers the basics of surds with easy and tricky examples to help you understand.