State the type of the decimal expansion of $\frac{1}{7}$.

Between two rational numbers,

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

Insert a rational number and an irrational number between the following:
$2.357$ and $3.121$

If $125^{x} = \frac{25}{5^{x}},$ then find the value of $x$.

Find the value of the following expression correct to three decimal places,rationalizing the denominator if needed. Take $\sqrt{2} = 1.414$,$\sqrt{3} = 1.732$,and $\sqrt{5} = 2.236$.
$\frac{4}{3 \sqrt{3}-2 \sqrt{2}} + \frac{3}{3 \sqrt{3}+2 \sqrt{2}}$