1. The problem is to rewrite the expression $\left(\frac{5}{8}\right)^{-2}$ without an exponent. 2. Recall the rule for negative exponents: $a^{-n} = \frac{1}{a^n}$. 3. Apply this rule to the expression: $$\left(\frac{5}{8}\right)^{-2} = \frac{1}{\left(\frac{5}{8}\right)^2}$$ 4. Next, square the fraction inside the denominator: $$\left(\frac{5}{8}\right)^2 = \frac{5^2}{8^2} = \frac{25}{64}$$ 5. Substitute back: $$\frac{1}{\frac{25}{64}}$$ 6. Dividing by a fraction is the same as multiplying by its reciprocal: $$1 \times \frac{64}{25} = \frac{64}{25}$$ 7. Therefore, the expression without an exponent is: $$\boxed{\frac{64}{25}}$$