## Powers in finite fields

For $$n\in \mathbb N$$ and an asociative operation $$\cdot$$ let $$a^n=a\cdot a\cdot \ldots \cdot a$$, where the element $$a$$ appears $$n$$ times in the product.
Determine values $$2^{101},3^{1\,001}$$ and $$4^{1\,000\,001}$$ in the field $$\mathbb Z_{17}$$.
Determine $$5^{100},8^{200},11^{300}$$ and $$18^{400}$$ in the field $$\mathbb Z_{19}$$.