1. **State the problem:** Simplify the expression $$\left(\frac{25}{4}\right)^{-\frac{3}{2}}$$ and express the answer with positive exponents. 2. **Recall the rule for negative exponents:** For any nonzero number $a$, $$a^{-n} = \frac{1}{a^n}$$ where $n$ is positive. 3. **Apply the negative exponent rule:** $$\left(\frac{25}{4}\right)^{-\frac{3}{2}} = \frac{1}{\left(\frac{25}{4}\right)^{\frac{3}{2}}}$$ 4. **Simplify the positive exponent:** $$\left(\frac{25}{4}\right)^{\frac{3}{2}} = \left(\left(\frac{25}{4}\right)^{\frac{1}{2}}\right)^3$$ 5. **Calculate the square root:** $$\left(\frac{25}{4}\right)^{\frac{1}{2}} = \frac{\sqrt{25}}{\sqrt{4}} = \frac{5}{2}$$ 6. **Raise to the power 3:** $$\left(\frac{5}{2}\right)^3 = \frac{5^3}{2^3} = \frac{125}{8}$$ 7. **Substitute back into the expression:** $$\frac{1}{\left(\frac{25}{4}\right)^{\frac{3}{2}}} = \frac{1}{\frac{125}{8}}$$ 8. **Simplify the division of fractions:** $$\frac{1}{\frac{125}{8}} = \frac{1 \times 8}{125} = \frac{8}{125}$$ **Final answer:** $$\boxed{\frac{8}{125}}$$