Simplifying Nested Radicals

The problem requires simplifying the expression $\frac{\sqrt{5+2\sqrt{6}}}{\sqrt{5-2\sqrt{6}}}$. This involves simplifying nested square roots of the form $\sqrt{a \pm 2\sqrt{b}}$.

Step 1: Simplify Nested Radicals

To simplify $\sqrt{a \pm 2\sqrt{b}}$, we look for two numbers, $x$ and $y$, such that $x+y = a$ and $x \times y = b$. If such numbers are found, then $\sqrt{a \pm 2\sqrt{b}} = \sqrt{x} \pm \sqrt{y}$ (assuming $\sqrt{x} > \sqrt{y}$ for the minus case).

Step 2: Simplify Numerator $\boldsymbol{\sqrt{5+2\sqrt{6}}}$

For $\sqrt{5+2\sqrt{6}}$, we have $a=5$ and $b=6$. We need $x+y=5$ and $x \times y = 6$. The numbers are $x=3$ and $y=2$. Thus, $\sqrt{5+2\sqrt{6}} = \sqrt{3} + \sqrt{2}$.

Step 3: Simplify Denominator $\boldsymbol{\sqrt{5-2\sqrt{6}}}$

For $\sqrt{5-2\sqrt{6}}$, we use the same $x=3$ and $y=2$. Since $\sqrt{3} > \sqrt{2}$, we have $\sqrt{5-2\sqrt{6}} = \sqrt{3} - \sqrt{2}$.

Step 4: Substitute Simplified Terms

Substitute the simplified terms back into the original expression:

$ \frac{\sqrt{5+2\sqrt{6}}}{\sqrt{5-2\sqrt{6}}} = \frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}} $

Step 5: Rationalize the Denominator

Multiply the numerator and the denominator by the conjugate of the denominator, which is $(\sqrt{3}+\sqrt{2})$:

$ \frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}-\sqrt{2}} \times \frac{\sqrt{3}+\sqrt{2}}{\sqrt{3}+\sqrt{2}} $

This simplifies to:

$ \frac{(\sqrt{3}+\sqrt{2})^2}{(\sqrt{3})^2 - (\sqrt{2})^2} $

Expand the numerator and simplify the denominator:

$ \frac{(\sqrt{3})^2 + (\sqrt{2})^2 + 2(\sqrt{3})(\sqrt{2})}{3 - 2} = \frac{3 + 2 + 2\sqrt{6}}{1} $

The final simplified expression is:

$ 5 + 2\sqrt{6} $