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

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 $\sqrt{5} = 2.236$.
$\frac{4}{3 \sqrt{3}-2 \sqrt{2}} + \frac{3}{3 \sqrt{3}+2 \sqrt{2}}$

Insert a rational number and an irrational number between the following:
$0$ and $0.1$

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

Find three rational numbers between $\frac{2}{3}$ and $\frac{4}{5}$.

Prove that $\left(\frac{x^{a}}{x^{b}}\right)^{a+b} \times \left(\frac{x^{b}}{x^{c}}\right)^{b+c} \times \left(\frac{x^{c}}{x^{a}}\right)^{c+a} = 1$.