1. Let's start by stating the problem: We want to understand the next phase of surds, which involves simplifying expressions containing roots (square roots, cube roots, etc.). 2. The key formula to remember is that surds can be simplified by factoring the number inside the root into perfect powers. For example, $$\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$$. 3. Important rules: - You can multiply and divide surds if they have the same root. - You can simplify surds by extracting perfect squares (or cubes, etc.) from under the root. 4. Example: Simplify $$\sqrt{50}$$. - Factor 50 as $$25 \times 2$$. - Apply the rule: $$\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2}$$. - Since $$\sqrt{25} = 5$$, the expression simplifies to $$5\sqrt{2}$$. 5. Another example: Simplify $$\sqrt{72}$$. - Factor 72 as $$36 \times 2$$. - Apply the rule: $$\sqrt{72} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}$$. 6. For cube roots, the same principle applies. For example, $$\sqrt[3]{54}$$: - Factor 54 as $$27 \times 2$$. - $$\sqrt[3]{54} = \sqrt[3]{27} \times \sqrt[3]{2} = 3\sqrt[3]{2}$$. 7. Summary: To simplify surds, factor the number inside the root into perfect powers and extract them outside the root. This is the next phase of surds: mastering simplification and manipulation of roots.