1. **State the problem:** Simplify the expression $$(32x^{20}y^{30})^{-\frac{3}{5}}$$ to its simplest form using only positive exponents. 2. **Recall the rule for fractional exponents:** For any base $a$ and rational exponent $m/n$, $$(a^{m/n}) = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$$. 3. **Recall the rule for negative exponents:** $$a^{-m} = \frac{1}{a^m}$$. 4. **Apply the negative exponent rule:** $$ (32x^{20}y^{30})^{-\frac{3}{5}} = \frac{1}{(32x^{20}y^{30})^{\frac{3}{5}}} $$ 5. **Rewrite the base inside the denominator:** $$ (32x^{20}y^{30})^{\frac{3}{5}} = 32^{\frac{3}{5}} \cdot (x^{20})^{\frac{3}{5}} \cdot (y^{30})^{\frac{3}{5}} $$ 6. **Simplify each term using the power of a power rule:** $$ 32^{\frac{3}{5}}, \quad x^{20 \cdot \frac{3}{5}} = x^{12}, \quad y^{30 \cdot \frac{3}{5}} = y^{18} $$ 7. **Simplify $32^{\frac{3}{5}}$:** Since $32 = 2^5$, $$ 32^{\frac{3}{5}} = (2^5)^{\frac{3}{5}} = 2^{5 \cdot \frac{3}{5}} = 2^3 = 8 $$ 8. **Combine all simplified terms:** $$ (32x^{20}y^{30})^{\frac{3}{5}} = 8x^{12}y^{18} $$ 9. **Write the final simplified expression:** $$ (32x^{20}y^{30})^{-\frac{3}{5}} = \frac{1}{8x^{12}y^{18}} $$ This expression uses only positive exponents and is fully simplified.