1. Let's start by stating the problem: solving radicals means simplifying expressions that contain roots, such as square roots, cube roots, etc. 2. There are different types of radicals to consider: - Simple radicals like $\sqrt{a}$ - Radicals with denominators like $\frac{1}{\sqrt{b}}$ - Radicals with radical denominators like $\frac{1}{\sqrt{c}}$ 3. Important rules: - $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$ - To simplify radicals, factor the number inside the root into perfect squares. - To rationalize denominators, multiply numerator and denominator by the radical in the denominator. 4. Example 1: Simplify $\sqrt{50}$ - Factor 50 as $25 \times 2$ - $\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2}$ 5. Example 2: Simplify $\frac{1}{\sqrt{3}}$ - Multiply numerator and denominator by $\sqrt{3}$ to rationalize denominator: - $\frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{\cancel{\sqrt{3}} \times \cancel{\sqrt{3}}} = \frac{\sqrt{3}}{3}$ 6. Example 3: Simplify $\frac{5}{\sqrt{2} + 1}$ - Multiply numerator and denominator by the conjugate $\sqrt{2} - 1$: - $\frac{5}{\sqrt{2} + 1} \times \frac{\sqrt{2} - 1}{\sqrt{2} - 1} = \frac{5(\sqrt{2} - 1)}{(\sqrt{2} + 1)(\sqrt{2} - 1)}$ - Denominator simplifies using difference of squares: - $(\sqrt{2})^2 - 1^2 = 2 - 1 = 1$ - So expression becomes $5(\sqrt{2} - 1) = 5\sqrt{2} - 5$ 7. Summary: - Simplify radicals by factoring inside the root. - Rationalize denominators by multiplying numerator and denominator by the radical or conjugate. Final answers: - $\sqrt{50} = 5\sqrt{2}$ - $\frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$ - $\frac{5}{\sqrt{2} + 1} = 5\sqrt{2} - 5$