Express $0.\overline{4}$ in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$.

For each question,select the proper option from four options given,to make the statement true: $(5^{\frac{3}{4}})^{\frac{4}{3}} = \dots$

Rationalise the denominator in each of the following and hence evaluate by taking $\sqrt{2}=1.414, \sqrt{3}=1.732$ and $\sqrt{5}=2.236,$ up to three decimal places.
$\frac{1}{\sqrt{3}+\sqrt{2}}$

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

For each question,select the proper option from four options given,to make the statement true: The rationalising factor of $4-\sqrt{5}$ is...........

If $\frac{\sqrt{7}+\sqrt{5}}{\sqrt{7}-\sqrt{5}}=a+b \sqrt{35},$ find the value of $a$ and $b$.