Simplifying Nested Radicals (Double Radicals)
Big Idea: A nested radical (or double radical) is a radical expression that contains another radical inside it, such as $\sqrt{a + \sqrt{b}}$. Our goal is to "un-nest" these into a simpler form like $\sqrt{m} + \sqrt{n}$.
1. What Are Nested Radicals?
A nested radical looks like this:
$$\sqrt{a \pm \sqrt{b}}$$
For example: $\sqrt{5 + 2\sqrt{6}}$ or $\sqrt{7 - 4\sqrt{3}}$
These expressions look complicated, but many of them can be simplified into a much cleaner form. The key insight is that some nested radicals are actually perfect squares in disguise!