# Exponents in 2015

Algebra Level 5

Let $$a_1,$$ $$a_2,$$ $$a_3,$$ $$a_4$$ be complex numbers such that

\begin{align*} a_1 + a_2 + a_3 + a_4 &= 2 \\ a_1^2 + a_2^2 + a_3^2 + a_4^2 &= 0 \\ a_1^3 + a_2^3 + a_3^3 + a_4^3 &= 1 \\ a_1^4 + a_2^4 + a_3^4 + a_4^4 &= 5. \end{align*}

If $$a_1^6 + a_2^6 + a_3^6 + a_4^6 = \frac{m}{n}$$ for some relatively prime positive integers $$m$$ and $$n,$$ find $$m + n.$$

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