Express the following in the form $\frac{p}{q},$ where $p$ and $q$ are integers and $q \neq 0$ :
$0.1\overline{34}$

Classify the following numbers as rational or irrational with justification:
$(i)$ $-\sqrt{0.4}$
$(ii)$ $\frac{\sqrt{12}}{\sqrt{75}}$

Find the value of the following expression correct to three decimal places,rationalizing the denominator if needed. Take $\sqrt{2} = 1.414$,$\sqrt{3} = 1.732$,and $\sqrt{5} = 2.236$.
$\frac{4}{3 \sqrt{3}-2 \sqrt{2}} + \frac{3}{3 \sqrt{3}+2 \sqrt{2}}$

Simplify: $\frac{11^{\frac{1}{3}}}{11^{\frac{1}{5}}}$

Rationalise the denominator of the following expression and evaluate it by taking $\sqrt{3} = 1.732$,up to three decimal places:
$\frac{4}{\sqrt{3}}$

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