Express the following in the form $\frac{p}{q},$ where $p$ and $q$ are integers and $q \neq 0$ :
$0 . \overline{001}$

Rationalise the denominator of the following expression:
$\frac{30}{5 \sqrt{3}-3 \sqrt{5}}$

Fill in the blank to make the following statement true:
$\sqrt[11]{1} = \ldots$

Find the value of the following expression correct to three decimal places,rationalizing the denominator if needed. Given $\sqrt{2} = 1.414$,$\sqrt{3} = 1.732$,and $\sqrt{5} = 2.236$.
$\frac{1}{\sqrt{3} + \sqrt{2}}$

Is $\sqrt{8^{2}+15^{2}}$ a rational number or an irrational number?

Simplify:
$\frac{9^{\frac{1}{3}} \times 27^{-\frac{1}{2}}}{3^{\frac{1}{6}} \times 3^{-\frac{2}{3}}}$