The problem is as follows:
First simplify:
$$x^{x^{-5}(5)}=5^{-\frac{25}{\sqrt[5]{5^{16}}}}$$
Then using $x$ find: $x^{-50}$
The alternatives in my book are as follows:
$\begin{array}{ll} 1.&5\\ 2.&\frac{1}{5}\\ 3.&25\\ 4.&125\\ \end{array}$
I'm confused about solving this problem, how should it be approached?
$x^{x^{-5}(5)}=5^{-\frac{25}{\sqrt[5]{5^{16}}}}$
$5^{-\frac{25}{\sqrt[5]{5^{16}}}}=5^{-\frac{1}{5\sqrt[5]{5}}}$
$5^{-\frac{1}{5\sqrt[5]{5}}}=5^{-\frac{1}{5^{\frac{6}{5}}}}$
But that's where I'm stuck. Where to go from there?. I can't seem to find a way to get that problematic $x$. Can someone help me with the radical manipulation?. Does this requires any additional multiplication?
Im slightly confused about your notation, do you mean: $$x^{5x^{-5}}$$ or: $$\left(x^{x^{-5}}\right)^5$$
Anyways, we can start with the other side, we have: $$5^{-\frac{25}{\sqrt[5]{5^{16}}}}\tag{1}$$ if we look at the power first: $$-\frac{25}{\sqrt[5]{5^{16}}}=-\frac{5^2}{(5^{16})^{1/5}}=-\frac{5^2}{5^{16/5}}=-5^{2-16/5}=-5^{-6/5}$$ so we can simplify $(1)$ to: $$5^{-5^{-6/5}}$$ now equate the LHS to this