1. **State the problem:** Simplify the expression $$\left(\frac{8}{5}\right)^{-1}$$. 2. **Recall the rule for negative exponents:** For any nonzero number $x$ and integer $n$, $$x^{-n} = \frac{1}{x^n}$$. 3. **Apply the rule:** $$\left(\frac{8}{5}\right)^{-1} = \frac{1}{\left(\frac{8}{5}\right)^1}$$ 4. **Simplify the expression:** $$\frac{1}{\frac{8}{5}} = \frac{1 \times 5}{8} = \frac{5}{8}$$ 5. **Final answer:** $$\boxed{\frac{5}{8}}$$