Divide $12 \sqrt{30}$ by $3 \sqrt{5}$.

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

Express the following in the form $\frac{p}{q},$ where $p$ and $q$ are integers and $q \neq 0$ :
$0.404040 \ldots$

State whether the following statement is true or false:
$\sqrt{49}$ is an irrational number.

Insert a rational number and an irrational number between the following:
$0.0001$ and $0.001$

Find which of the variables $x, y, z$ and $u$ represent rational numbers and which irrational numbers:
$(i)$ $x^{2}=5$
$(ii)$ $y^{2}=9$
$(iii)$ $z^{2}=0.04$
$(iv)$ $u^{2}=\frac{17}{4}$