Simplify: $\frac{7 \sqrt{3}}{\sqrt{10}+\sqrt{3}}-\frac{2 \sqrt{5}}{\sqrt{6}+\sqrt{5}}-\frac{3 \sqrt{2}}{\sqrt{15}+3 \sqrt{2}}$

If $a = \frac{3+\sqrt{5}}{2}$,then find the value of $a^{2} + \frac{1}{a^{2}}$.

Write the number in short form: $3.8232323 \ldots$

Calculate the sum: $0. \overline{3} + 0.4 \overline{7}$

State whether the following statements are true or false. Justify your answer.
$(i)$ The number of rational numbers between $15$ and $18$ is finite.
$(ii)$ There are numbers which cannot be written in the form $\frac{p}{q}$,where $q \neq 0$ and $p, q$ are both integers.

Find the value of $b$ :
$\frac{\sqrt{2}+\sqrt{3}}{3 \sqrt{2}-2 \sqrt{3}}=2-b \sqrt{6}$