Method and WIP please ^^
$\displaystyle \frac{1}{8\sqrt{8}}$
Give answer in the form $\displaystyle \frac{\sqrt{2}}{p}$ Where p is a positive interger
For $\displaystyle \sqrt{8} = 2\sqrt{2}$ you need to know that $\displaystyle \sqrt{a^2} = a$ and $\displaystyle \sqrt{ab} = \sqrt{a}\sqrt{b}$
$\displaystyle 8 = 2^3 = 2^2 \cdot 2$
$\displaystyle \sqrt{8} = \sqrt{2^2 \cdot 2} = \sqrt{2^2} \cdot \sqrt{2} = 2\sqrt{2}$
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The $\displaystyle \frac{\sqrt{2}}{\sqrt{2}}$ comes about because we can multiply a fraction by 1 and leave it unchanged.
$\displaystyle \frac{\sqrt{2}}{\sqrt{2}} = 1$
Here's a more obvious way...
$\displaystyle \frac{1}{8\sqrt{8}}=\frac{\sqrt{2}}{\sqrt{2}}\ \frac{1}{8\sqrt{8}}=\frac{\sqrt{2}}{8\sqrt{2}\sqrt {8}}$
$\displaystyle =\frac{\sqrt{2}}{8\sqrt{2(8)}}=\frac{\sqrt{2}}{8\s qrt{16}}$
$\displaystyle =\frac{\sqrt{2}}{8(4)}=\frac{\sqrt{2}}{32}$