Dealing with Surds and Reciprocals

Question

If $a= (5-\sqrt5)/(5+\sqrt5)$, evaluate, showing working:

1. $a+\frac{1}{a}$
2. $a^2+\frac{1}{a^2}$

I attempted these questions but got them wrong, the answer for part (1) is $3$, I got $2/5$.

My working:

\begin{align} a+1/a &= \frac{5-\sqrt5}{5+\sqrt5} + \frac{5+\sqrt5}{5-\sqrt5}\\ &=\frac{(5-\sqrt5)^2 + (5+\sqrt5)^2}{(5+\sqrt5)(5+\sqrt5)} \quad &\text{Common denominator}\\ &=\frac{(25-10\sqrt5 -5)+(25+10\sqrt5+5)}{25-5} \quad &\text{Expanding the brackets}\\ &=\frac{50}{20}\\ &=\frac{5}{2}\\ \end{align}

I've triple checked my working out, and can't find any algebraic mistakes. How is the answer $3$?

For question $2$, how is it done? I'm still a bit confused. Can the $a^2 +1/a^2$ be changed to something to do with part $1$?

Answer 1

You've made a mistake while expanding $(5-\sqrt5)^2$. It's $25-10\sqrt{5}+5.$ For the second part, use the fact that$a^{2}+\frac{1}{a^2}=(a+\frac{1}{a})^2-2$.

Answer 2

After getting a common denominator, you have $25 -20\sqrt{5}-5$ as the start of the numerator. The last term I wrote there should be $+5$, not $-5$.