<s>[INST] negative exponent problem $$\sqrt{\frac{1}{3^0 + 3^{-1} + 3^{-2} + 3^{-3} + 3^{-4}}}$$
Does this equal =
$$
\begin{align*}
& \sqrt{3^0 + 3^1 + 3^2 + 3^3 + 3^4} \\
=&\sqrt{1 + 3 + 9 + 27 + 81} \\
=&\sqrt{121} \\
=&11. 
\end{align*}
$$
The answer is apparently $\frac{9}{11}$ and I'm not sure what rule of negative exponents I got wrong.
The rule I'm using, incorrectly, is this:
$$\frac{1}{3^{-2}} = 3^2 = 9.$$
 [/INST] When you have a doubt with such expressions (I use here you specific problem), inside the square root, multiply both numerator and denominator by the reverse of the most negative power (here : 3^4) and perform the multiplications by (3^4) for each term of the denominator. Then, perform the additions and simplifications.
</s>