2. (a) Simplify $$\sqrt{32} + \sqrt{18}$$ giving your answer in the form $a\sqrt{2}$, where $a$ is an integer - Edexcel - A-Level Maths Pure - Question 4 - 2012 - Paper 1

To simplify the expression, first substitute the simplified forms from part (a):
$\sqrt{32} + \sqrt{18} = 7\sqrt{2}$.

Thus, the expression becomes:

$\frac{7\sqrt{2}}{3 + \sqrt{2}}$.

Next, we rationalize the denominator:

1. Multiply numerator and denominator by the conjugate:
   $\frac{7\sqrt{2}(3 - \sqrt{2})}{(3 + \sqrt{2})(3 - \sqrt{2})}$.

2. Calculate the denominator:
   $(3 + \sqrt{2})(3 - \sqrt{2}) = 3^2 - (\sqrt{2})^2 = 9 - 2 = 7.$

3. Calculate the numerator:
   $7\sqrt{2}(3 - \sqrt{2}) = 21\sqrt{2} - 7(2) = 21\sqrt{2} - 14.$

Now we have:

$\frac{21\sqrt{2} - 14}{7}$.

4. Simplify:
   $\frac{21\sqrt{2}}{7} - \frac{14}{7} = 3\sqrt{2} - 2.$

Thus, the final answer for part (b) is $3\sqrt{2} - 2$.