The goal is to simplify the given mathematical expression:

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

Simplifying Nested Square Roots

We need to simplify the terms $\sqrt{5+2\sqrt{6}}$ and $\sqrt{5-2\sqrt{6}}$. We look for two numbers whose sum is 5 and product is 6. These numbers are 3 and 2.

• Using the identity $\sqrt{a+2\sqrt{b}} = \sqrt{x} + \sqrt{y}$ where $x+y=a$ and $xy=b$:

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

• Using the identity $\sqrt{a-2\sqrt{b}} = \sqrt{x} - \sqrt{y}$ (where $x>y$):

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

Substituting Simplified Terms

Now substitute these simplified terms back into the numerator of the original expression:

$ \text{Numerator} = (\sqrt{3} + \sqrt{2}) + (\sqrt{3} - \sqrt{2}) $

Combine like terms:

$ \text{Numerator} = \sqrt{3} + \sqrt{2} + \sqrt{3} - \sqrt{2} = 2\sqrt{3} $

Final Simplification

Now, divide the simplified numerator by the denominator ($\sqrt{2}$):

$ \frac{2\sqrt{3}}{\sqrt{2}} $

To simplify further, rationalize the denominator by multiplying the numerator and denominator by $\sqrt{2}$:

$ \frac{2\sqrt{3}}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{2\sqrt{3 \times 2}}{\sqrt{2 \times 2}} = \frac{2\sqrt{6}}{2} $

Cancel out the common factor of 2:

$ \sqrt{6} $

Therefore, the simplified expression is $\sqrt{6}$.