Show that $$ \sqrt{180} - 2\sqrt{5} $$ $$ \frac{\sqrt{5} - 5}{5} $$ can be written in the form $$ a + \frac{\sqrt{5}}{b} $$ where a and b are integers. - Edexcel - GCSE Maths - Question 21 - 2020 - Paper 1

Now, let's expand the numerator:

$(\sqrt{180} - 2\sqrt{5})(\sqrt{5} + 5) = \sqrt{180} \cdot \sqrt{5} + 5\sqrt{180} - 2\sqrt{5} \cdot \sqrt{5} - 10\sqrt{5}$

This simplifies to:

$\sqrt{900} + 5\sqrt{180} - 10 - 10\sqrt{5}$

Noting that $\sqrt{900} = 30$ and simplifying further, we can then separate terms containing $√5$.

So far, we have:

$\frac{(30 + 5\sqrt{180} - 10 - 10\sqrt{5})}{-20}$

This can be written as:

$\frac{30 - 10}{-20} + \frac{5\sqrt{180} - 10\sqrt{5}}{-20}$

which yields:

$-1 + \frac{5\sqrt{180} - 10\sqrt{5}}{-20}$

By factoring out the terms in the numerator, we achieve:

$-1 + \frac{\sqrt{5}(b)}{-20}$

Thus, confirming the presence of integer values for $a$ and $b$.

Combining all the steps, we conclude that:

$a = -1, b = -20$

Hence the given expression can be written in the desired form: $a + \frac{\sqrt{5}}{b}$ where $a$ and $b$ are integers.