Find the value of $a$ in the following :

$\frac{6}{3 \sqrt{2}-2 \sqrt{3}}=3 \sqrt{2}-a \sqrt{3}$

Simplify the following:

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

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

Insert a rational number and an irrational number between the following:

$6.375289$ and $6.375738$

Find three rational numbers between

$0.1$ and $0.11$

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

$0.5 \overline{7}$