Find the geometric mean (or mean proportional) of $\sqrt{2}$ and $\sqrt{8}$.

1. Problem Description

Find the geometric mean (or mean proportional) of $\sqrt{2}$ and $\sqrt{8}$.

2. Solution Steps

The geometric mean (or mean proportional) of two numbers $a$ and $b$ is a number $x$ such that $\frac{a}{x} = \frac{x}{b}$, or $x^2 = ab$. Thus, $x = \pm\sqrt{ab}$.

In our case, $a = \sqrt{2}$ and $b = \sqrt{8}$.

Therefore, the geometric mean is $x = \pm\sqrt{\sqrt{2} \cdot \sqrt{8}}$.

$x = \pm\sqrt{\sqrt{16}}$.

$x = \pm\sqrt{4}$.

$x = \pm 2$.

3. Final Answer

$\pm 2$