Simplify the following expression: $(\sqrt{15}+\sqrt{7})(\sqrt{15}-\sqrt{7})$

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

If $a = 5 + 2\sqrt{6}$ and $b = \frac{1}{a}$,then what will be the value of $a^2 + b^2$?

Simplify:
$\frac{9^{\frac{1}{3}} \times 27^{-\frac{1}{2}}}{3^{\frac{1}{6}} \times 3^{-\frac{2}{3}}}$

Express $0.\overline{27}$ in the $\frac{p}{q}$ form.

For each question,select the proper option from the four options given to make the statement true: $(\sqrt{5}+3)^{2}$ is a / an $\ldots \ldots \ldots$ number.