The value of $(-0.1)^{-0.1}$

Question

I saw a video about how the answer for this is complex because- $(-0.1)^{-0.1}$

$\frac{1}{(-0.1)^{0.1}}$

$\frac{1}{(-0.1)^\frac{1}{10}}$

$\frac{1}{\sqrt[10]{-0.1}}$

$\sqrt[10]{-0.1} \;\epsilon \;\mathbb{C}$

But if I write 1/10 as 2/20

$-0.1^\frac{2}{20}$

$\sqrt[20]{0.01}\;\epsilon\;\mathbb{R}$

Answer 1

ERROR ... You used the phony law $(a^b)^c = a^{(bc)}$. It is "phony"unless you check the conditions under which it holds: namely, $a>0$.

You wrote $\big((-0.01)^2\big)^{\frac{1}{20}}= (-0.01)^{(2\cdot\frac{1}{20})}$ withouth checking whether or not $-0.01 > 0$.

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