1. **State the problem:** Simplify the expression $$\frac{-4(r^4t^{-2})^{-1}}{4r^2t^4}$$ so that it contains only positive exponents. 2. **Recall the rules:** - When raising a power to another power, multiply the exponents: $$(a^m)^n = a^{mn}$$ - Negative exponents mean reciprocal: $$a^{-m} = \frac{1}{a^m}$$ - When dividing like bases, subtract exponents: $$\frac{a^m}{a^n} = a^{m-n}$$ 3. **Simplify the numerator:** $$ (r^4 t^{-2})^{-1} = r^{4 \times (-1)} t^{-2 \times (-1)} = r^{-4} t^{2} $$ 4. **Rewrite the entire fraction:** $$ \frac{-4 r^{-4} t^{2}}{4 r^{2} t^{4}} $$ 5. **Cancel common factors:** $$ \frac{\cancel{-4} r^{-4} t^{2}}{\cancel{4} r^{2} t^{4}} = \frac{r^{-4} t^{2}}{r^{2} t^{4}} $$ 6. **Apply division of like bases:** $$ r^{-4 - 2} t^{2 - 4} = r^{-6} t^{-2} $$ 7. **Rewrite with positive exponents:** $$ r^{-6} = \frac{1}{r^{6}}, \quad t^{-2} = \frac{1}{t^{2}} $$ 8. **Final simplified expression:** $$ \frac{1}{r^{6} t^{2}} $$ **Answer:** $$\frac{1}{r^{6} t^{2}}$$