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 3 - 2015 - Paper 1

To simplify [ \frac{\sqrt{2}}{2\sqrt{5} - 3\sqrt{2}} ], we start by rationalizing the denominator.

Multiply the numerator and denominator by the conjugate of the denominator, which is $2\sqrt{5} + 3\sqrt{2}$:

[ \frac{\sqrt{2}(2\sqrt{5} + 3\sqrt{2})}{(2\sqrt{5} - 3\sqrt{2})(2\sqrt{5} + 3\sqrt{2})} ]

Calculating the denominator:

$(2\sqrt{5})^2 - (3\sqrt{2})^2 = 4 \cdot 5 - 9 \cdot 2 = 20 - 18 = 2.$

Now for the numerator:

$\sqrt{2}(2\sqrt{5}) + \sqrt{2}(3\sqrt{2}) = 2\sqrt{10} + 3 \cdot 2 = 2\sqrt{10} + 6.$

Combining these results, we get:

[ \frac{2\sqrt{10} + 6}{2}. ]

Now, divide each term in the numerator by 2:

[ \sqrt{10} + 3. ]

Thus, the final answer can be expressed as ( 3 + \sqrt{10} ), where $a = 3$ and $b = 10$.