[tex]Sure, let's break down the problem step by step.
First, consider the expression given:
\[
\left( -\frac{8}{11} \right)^{-5} + \left( \left( -\frac{8}{11} \right)^2 \right)^3
\]
Let's simplify each part separately:
1. Simplify \(\left( -\frac{8}{11} \right)^{-5}\):
Using the property of exponents: \(a^{-n} = \frac{1}{a^n}\)
\[
\left( -\frac{8}{11} \right)^{-5} = \frac{1}{\left( -\frac{8}{11} \right)^5}
= \frac{1}{\left( -\frac{8}{11} \right)^5}
= \left( -\frac{11}{8} \right)^5 = - \left( \frac{11}{8} \right)^5
\]
2. Simplify \(\left( \left( -\frac{8}{11} \right)^2 \right)^3\):
Using the property of exponents: \((a^m)^n = a^{mn}\)
\[
\left( \left( -\frac{8}{11} \right)^2 \right)^3 = \left( -\frac{8}{11} \right)^{2 \cdot 3} = \left( -\frac{8}{11} \right)^6
= \left( \frac{8}{11} \right)^6
\]
So now, our expression becomes:
\[
-\left( \frac{11}{8} \right)^5 + \left( \frac{8}{11} \right)^6
\]
We need to find the reciprocal of this expression:
\[
\text{Reciprocal of}\left[ -\left( \frac{11}{8} \right)^5 + \left( \frac{8}{11} \right)^6 \right]
= \frac{1}{-\left( \frac{11}{8} \right)^5 + \left( \frac{8}{11} \right)^6}
\]
So the reciprocal of the given expression is:
\[
\boxed{\frac{1}{-\left( \frac{11}{8} \right)^5 + \left( \frac{8}{11} \right)^6}}
\][/tex]
