isimplify: \( \frac{\sqrt{2+2}+\sqrt{5}}{\sqrt{10}} \)

Let's simplify the expression \( \frac{\sqrt{2+2}+\sqrt{5}}{\sqrt{10}} \). First, simplify \( \sqrt{2+2} \): \[ \sqrt{2+2} = \sqrt{4} = 2 \] Now, substituting this back into the expression: \[ \frac{2 + \sqrt{5}}{\sqrt{10}} \] Next, we can multiply the numerator and denominator by \( \sqrt{10} \) to rewrite it: \[ \frac{2 + \sqrt{5}}{\sqrt{10}} = \frac{(2 + \sqrt{5})\sqrt{10}}{10} \] This expands to: \[ \frac{2\sqrt{10} + \sqrt{50}}{10} \] Since \( \sqrt{50} = 5\sqrt{2} \), we can rewrite the expression as: \[ \frac{2\sqrt{10} + 5\sqrt{2}}{10} \] Thus, the simplified result is: \[ \frac{2\sqrt{10}}{10} + \frac{5\sqrt{2}}{10} = \frac{\sqrt{10}}{5} + \frac{\sqrt{2}}{2} \] Therefore, the final simplified expression is: \(\frac{2\sqrt{10} + 5\sqrt{2}}{10}\).