Find the value of $a$ :

$\frac{5+2 \sqrt{3}}{7+4 \sqrt{3}}=a-6 \sqrt{3}$

Find the value

$\sqrt[5]{(243)^{-3}}$

Classify the following numbers as rational or irrational with justification:

$(i)$ $\sqrt{\frac{9}{27}}$

$(ii)$ $\frac{\sqrt{28}}{\sqrt{343}}$

Express $0.6+0 . \overline{7}+0.4 \overline{7}$ in the form $\frac{p}{q},$ where $p$ and $q$ are integers and $q \neq 0$.

Find which of the variables $x, y, z$ and $u$ represent rational numbers and which irrational numbers:

$(i)$ $x^{2}=5$

$(ii)$ $\quad y^{2}=9$

$(iii)$ $z^{2}=.04$

$(iv)$ $u^{2}=\frac{17}{4}$

The number obtained on rationalizing the denominator of $\frac{1}{7-\sqrt{2}}$ is