Is $0.3\overline{7}$ a rational number or an irrational number?

Express $0.5\overline{7}$ in the form $\frac{p}{q}$,where $p$ and $q$ are integers and $q \neq 0$.

Rationalise the denominator of the following:
$\frac{2}{3 \sqrt{3}}$

Classify the following numbers as rational or irrational with justification:
$(i)$ $\sqrt{\frac{9}{27}}$
$(ii)$ $\frac{\sqrt{28}}{\sqrt{343}}$

Classify the following numbers as rational or irrational with justification:
$(i)$ $\sqrt{196}$
$(ii)$ $3\sqrt{18}$

State whether the following statement is true or false. Justify your answer.
$\frac{\sqrt{15}}{\sqrt{3}}$ is written in the form $\frac{p}{q},$ where $q \neq 0,$ so it is a rational number.