Express $1.23 \overline{4}$ in the form $\frac{p}{q} ;$ where $p$ and $q$ are integers and $q \neq 0$

Find the values of each of the following correct to three places of decimals, rationalising the denominator if needed and taking $\sqrt{2}=1.414$ $\sqrt{3}=1.732$ and $\sqrt{5}=2.236$

$\frac{1}{\sqrt{3}+\sqrt{2}}$

Simplify $: \frac{(25)^{\frac{3}{2}} \times(243)^{\frac{3}{5}}}{(16)^{\frac{5}{4}} \times(8)^{\frac{4}{3}}}$

Find the values of each of the following correct to three places of decimals, rationalising the denominator if needed and taking $\sqrt{2}=1.414$ $\sqrt{3}=1.732$ and $\sqrt{5}=2.236$

$\frac{\sqrt{2}}{2+\sqrt{2}}$

Rationalise the denominator in each of the following

$\frac{5-2 \sqrt{6}}{5+2 \sqrt{6}}$

If $125^{x}=\frac{25}{5^{x}},$ then find the value of $x$.