Simplify the following:

$\sqrt[4]{81}-8 \sqrt[3]{216}+15 \sqrt[5]{32}+\sqrt{225}$

Find three different irrational numbers between the irrational numbers $\sqrt{2}$ and $\sqrt{5}$.

Simplify the following:

$\frac{3}{\sqrt{8}}+\frac{1}{\sqrt{2}}$

Simplify:

${{(625)^{-\frac{1}{2}}}^{-\frac{1}{4}}}^{2}$

Classify the following numbers as rational or irrational with justification:

$(i)$ $\sqrt{\frac{9}{27}}$

$(ii)$ $\frac{\sqrt{28}}{\sqrt{343}}$

If $x=5+2 \sqrt{6},$ then find the value of $x^{2}+\frac{1}{x^{2}}$ and $x^{3}+\frac{1}{x^{3}}$