1. Let's start by understanding what nested surds are. A nested surd is a surd (an expression containing a square root) inside another surd, for example, $\sqrt{3 + \sqrt{5}}$. 2. The goal is often to simplify nested surds into a simpler form without nested roots, if possible. 3. A common method is to assume the nested surd can be expressed as a sum or difference of simpler surds. For example, assume: $$\sqrt{3 + \sqrt{5}} = \sqrt{a} + \sqrt{b}$$ where $a$ and $b$ are positive numbers to be found. 4. Square both sides: $$3 + \sqrt{5} = (\sqrt{a} + \sqrt{b})^2 = a + b + 2\sqrt{ab}$$ 5. Equate the rational and irrational parts: - Rational part: $a + b = 3$ - Irrational part: $2\sqrt{ab} = \sqrt{5}$ 6. From the irrational part, square both sides: $$4ab = 5 \implies ab = \frac{5}{4}$$ 7. Now solve the system: $$a + b = 3$$ $$ab = \frac{5}{4}$$ 8. Treating $a$ and $b$ as roots of the quadratic equation: $$x^2 - 3x + \frac{5}{4} = 0$$ 9. Calculate the discriminant: $$\Delta = 3^2 - 4 \times 1 \times \frac{5}{4} = 9 - 5 = 4$$ 10. Find the roots: $$x = \frac{3 \pm \sqrt{4}}{2} = \frac{3 \pm 2}{2}$$ So, $$x_1 = \frac{5}{2} = 2.5, \quad x_2 = \frac{1}{2} = 0.5$$ 11. Therefore, $a = 2.5$ and $b = 0.5$ (or vice versa). 12. Substitute back: $$\sqrt{3 + \sqrt{5}} = \sqrt{2.5} + \sqrt{0.5} = \sqrt{\frac{5}{2}} + \sqrt{\frac{1}{2}} = \frac{\sqrt{5}}{\sqrt{2}} + \frac{1}{\sqrt{2}} = \frac{\sqrt{5} + 1}{\sqrt{2}}$$ 13. To rationalize the denominator: $$\frac{\sqrt{5} + 1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{(\sqrt{5} + 1)\sqrt{2}}{2}$$ 14. Final simplified form: $$\sqrt{3 + \sqrt{5}} = \frac{(\sqrt{5} + 1)\sqrt{2}}{2}$$ This is a simplified expression without nested surds. Nested surds can often be simplified by expressing them as sums of simpler surds and solving for unknowns using algebraic methods.