18 (a) Express $\sqrt{3} + \sqrt{12}$ in the form $a\sqrt{3}$ where $a$ is an integer - Edexcel - GCSE Maths - Question 19 - 2019 - Paper 1

We start by simplifying the expression $\left( \frac{1}{\sqrt{3}} \right)^{7}$:

$\left( \frac{1}{\sqrt{3}} \right)^{7} = \frac{1^{7}}{(\sqrt{3})^{7}} = \frac{1}{\sqrt{3^{7}}}.$

Next, we simplify $\sqrt{3^{7}}$:

$\sqrt{3^{7}} = \sqrt{3^{6} \cdot 3} = \sqrt{(3^{3})^{2} \cdot 3} = 3^{3} \sqrt{3} = 27\sqrt{3}.$

Thus, the expression becomes:

$\frac{1}{27\sqrt{3}}.$

To match the form $\frac{\sqrt{b}}{c}$, we multiply the numerator and denominator by $\sqrt{3}$:

$\frac{\sqrt{3}}{27 \cdot 3} = \frac{\sqrt{3}}{81}.$

In this case, $b = 3$ and $c = 81.$