Simplify the following:
$4 \sqrt{28} \div 3 \sqrt{7}$

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

Fill in the blank to make the following statement true:
$\sqrt[11]{1} = \ldots$

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

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 $\sqrt{5} = 2.236$.
$\frac{\sqrt{2}}{2+\sqrt{2}}$

Show that $0.\overline{076923} = \frac{1}{13}$.