<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] no you misses  $1$  from of division,it is  equal 
$\sqrt{1/(1+1/3+1/9+1/27+1/81)}$
please find now least common  multiple and simplify equation
just to simplify situation,least common multiple is $243$,we got it by $(81*27)/9$,
there $9$ is greatest common divisor,now let us solve it we would have
$(243+81+27+9+3)/243=363/243$ 
so we have $\sqrt {(1/(363/243)}$  or $\sqrt{243/363}$, we have common $3$,we we would have $\sqrt{81/121}$  and this is equal $9/11$
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