Simplify the following:
$\sqrt{45}-3 \sqrt{20}+4 \sqrt{5}$

Classify the following numbers as rational or irrational with justification:
$(i)$ $10.124124 \ldots$
$(ii)$ $1.010010001 \ldots$

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

If $x = 2 + \sqrt{3},$ then find the value of $x^{2} + \frac{1}{x^{2}}$ and $x^{3} + \frac{1}{x^{3}}.$

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

Rationalise the denominator in each of the following:
$\frac{3+2 \sqrt{2}}{3-2 \sqrt{2}}$