Let $a$ and $b$ be any positive real numbers. Determine the value of the expression

$$
\frac{\sqrt{\frac{a b}{2}}+\sqrt{8}}{\sqrt{\frac{a b+16}{8}+\sqrt{a b}}}
$$
Solution

Let $x=\sqrt{\frac{a b}{2}}+\sqrt{8}$. Then:

$$
\begin{aligned}
x^{2} & =\frac{a b}{2}+4 \sqrt{a b}+8 \\
& =4\left(\frac{a b+16}{8}+\sqrt{a b}\right) \\
& =4\left(\sqrt{\frac{a b+16}{8}+\sqrt{a b}}\right)^{2}
\end{aligned}
$$

Thus, the value of the desired expression is

$$
\begin{aligned}
\frac{\sqrt{\frac{a b}{2}}+\sqrt{8}}{\sqrt{\frac{a b+16}{8}+\sqrt{a b}}} & =\frac{x}{x / 2} \\
& =2
\end{aligned}
$$