Simplify (a) $(2 \sqrt{5})^2$ (b) $\frac{\sqrt{2}}{2\sqrt{5} - 3\sqrt{2}}$ giving your answer in the form $a + \sqrt{b}$, where $a$ and $b$ are integers. - Edexcel - A-Level Maths Pure - Question 5 - 2015 - Paper 1

To simplify the expression $\frac{\sqrt{2}}{2\sqrt{5} - 3\sqrt{2}}$, we first multiply the numerator and denominator by the conjugate of the denominator:

$$\frac{\sqrt{2}}{2\sqrt{5} - 3\sqrt{2}} \cdot \frac{2\sqrt{5} + 3\sqrt{2}}{2\sqrt{5} + 3\sqrt{2}}.$$

This results in:

$$\frac{\sqrt{2}(2\sqrt{5} + 3\sqrt{2})}{(2\sqrt{5})^2 - (3\sqrt{2})^2}.$$

We know that $(2\sqrt{5})^2 = 4 \cdot 5 = 20$ and $(3\sqrt{2})^2 = 9 \cdot 2 = 18$. Therefore, the denominator simplifies to:

$$20 - 18 = 2.$$

Now, substituting back, we get:

$$\frac{\sqrt{2}(2\sqrt{5} + 3\sqrt{2})}{2} = \frac{1}{2} \cdot \sqrt{2}(2\sqrt{5} + 3\sqrt{2}).$$

Distributing the $\sqrt{2}$ gives:

$$\frac{\sqrt{2} \cdot 2\sqrt{5}}{2} + \frac{3 \cdot 2}{2} = \sqrt{10} + 3.$$

Thus, the answer for part (b) is $3 + \sqrt{10}$, where $a=3$ and $b=10$.