Multiply $5 \sqrt{3}$ and $4 \sqrt{12}$.

If $x = 7 - 4\sqrt{3}$,then find the value of $x^{2} + \frac{1}{x^{2}}$.

Let $x$ be a rational number and $y$ be an irrational number. Is $xy$ necessarily irrational? Justify your answer with an example.

For each question,select the proper option from four options given,to make the statement true: $\frac{\sqrt{50}}{\sqrt{98}}$ is a $\ldots \ldots \ldots$ number.

$\sqrt[4]{\sqrt[3]{2^{2}}}$ equals

Simplify: $5 \sqrt{2} + 2 \sqrt{8} - 3 \sqrt{32} + 4 \sqrt{128}$ (in $\sqrt{2}$)