## Task A-1.1. (4 points)

Let $n$ be a natural number and $a \neq 0$ a real number. Completely simplify the fraction

$$
\frac{a^{3 n+1}-a^{4}}{a^{2 n+3}+a^{n+4}+a^{5}}
$$
Solution.

$$
\begin{aligned}
\frac{a^{3 n+1}-a^{4}}{a^{2 n+3}+a^{n+4}+a^{5}} & =\frac{a\left(a^{3 n}-a^{3}\right)}{a^{3}\left(a^{2 n}+a^{n+1}+a^{2}\right)} \\
& =\frac{a\left(a^{n}-a\right)\left(a^{2 n}+a^{n+1}+a^{2}\right)}{a^{3}\left(a^{2 n}+a^{n+1}+a^{2}\right)} \\
& =\frac{a\left(a^{n}-a\right)}{a^{3}}=\frac{a^{n-1}-1}{a}
\end{aligned}
$$

Note. The final solution can also be written in the form $a^{n-2}-a^{-1}$.