Simplify Expression E02202

1. **State the problem:** Simplify the expression $$\frac{2(p^{2})^{-4}}{2r^{-2}}$$ to its simplest form. 2. **Recall the rules:** - Negative exponents mean reciprocal: $$a^{-n} = \frac{1}{a^{n}}$$. - When dividing, you can cancel common factors. 3. **Rewrite the expression:** $$\frac{2(p^{2})^{-4}}{2r^{-2}} = \frac{2 \cdot p^{-8}}{2 \cdot r^{-2}}$$ because $$(p^{2})^{-4} = p^{2 \times (-4)} = p^{-8}$$. 4. **Cancel the common factor 2:** $$\frac{\cancel{2} \cdot p^{-8}}{\cancel{2} \cdot r^{-2}} = \frac{p^{-8}}{r^{-2}}$$ 5. **Rewrite the division of negative exponents:** $$\frac{p^{-8}}{r^{-2}} = p^{-8} \times r^{2}$$ because dividing by $r^{-2}$ is multiplying by $r^{2}$. 6. **Final simplified form:** $$r^{2} p^{-8}$$ or equivalently $$\frac{r^{2}}{p^{8}}$$. **Answer:** $$\frac{r^{2}}{p^{8}}$$