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

Let $x$ be a rational number and $y$ be an irrational number. Is $xy$ necessarily irrational? Justify your answer with an example.

Express $1.23\overline{4}$ in the form $\frac{p}{q}$,where $p$ and $q$ are integers and $q \neq 0$.

Show that $0.142857142857... = 0.\overline{142857} = \frac{1}{7}$.

Are there two irrational numbers whose sum and product both are rational? Justify.

State whether the following statements are true or false. Justify your answer.
$(i)$ The square of an irrational number is always rational.
$(ii)$ $\frac{\sqrt{12}}{\sqrt{3}}$ is not a rational number as $\sqrt{12}$ and $\sqrt{3}$ are not integers.