<s>[INST] Is $((\frac{1}{a})^{\frac{1}{b}})^{\frac{1}{c}}=\frac{1}{\sqrt[bc]{a}}$? Is $((\frac{1}{a})^{\frac{1}{b}})^{\frac{1}{c}}=\frac{1}{\sqrt[bc]{a}}$?
I believe it is since $((\frac{1}{a})^{\frac{1}{b}})^{\frac{1}{c}}=\frac{1}{\sqrt[b]{a}}^\frac{1}{c}=\frac{1}{\sqrt[bc]{a}}$.
Can someone please confirm my suspicions as I just came across a question that states in the answer that $((\frac{1}{a})^{\frac{1}{b}})^{\frac{1}{c}}=\sqrt[bc]{a}$ and I believe that this is a typo.
 [/INST] ${ \left( {\frac{1}{a}}^{\frac{1}{b}} \right)}^{\frac{1}{c}}= { \frac{1}{a}}^{ \frac{1}{b} \cdot \frac{1}{c}} =   { \frac{1}{a} }^{ \frac{1}{bc}} = \frac{1}{a^{\frac{1}{bc}}} = \frac{1}{\sqrt[bc]{a}}$
Thus, yes (assuming $a,b,c$ are all positive reals).
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