1. Consider the number $a=\sqrt{7+\sqrt{13}}-\sqrt{7-\sqrt{13}}$.

a) Prove that the number $a^{2}$ is a natural number.

b) Calculate $\left(a^{3}-2 a-1\right)^{2016}$.
1.a) $a^{2}=(\sqrt{7+\sqrt{13}}-\sqrt{7-\sqrt{13}})^{2}=7+\sqrt{13}-2 \cdot \sqrt{7+\sqrt{13}} \cdot \sqrt{7-\sqrt{13}}+7-\sqrt{13}=\ldots .1$ point $=14-2 \cdot \sqrt{7^{2}-\sqrt{13}^{2}}=14-2 \cdot 6=2 \in \mathbb{N}$............................................................ 2 points

b) $a^{3}-2 \cdot a=a \cdot\left(a^{2}-2\right)=a \cdot 0=0$....................................................................... 2 points

So $\left(a^{3}-2 \cdot a-1\right)^{2016}=(-1)^{2016}=1$................................................................. 2 points