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

Rationalise the denominator in each of the following

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

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 $\sqrt{5}=2.236$

$\frac{4}{3 \sqrt{3}-2 \sqrt{2}}+\frac{3}{3 \sqrt{3}+2 \sqrt{2}}$

Which of the following is not equal to $\left[\left(\frac{5}{6}\right)^{\frac{1}{5}}\right]^{-\frac{1}{6}} ?$

Find three rational numbers between

$\frac{1}{4}$ and $\frac{1}{5}$

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