1. **State the problem:** Given the equation $5p^{-3} = 8 \times 5^{-2}$, find the value of $p$. 2. **Recall the rules:** - Negative exponents mean reciprocal: $a^{-n} = \frac{1}{a^n}$. - When multiplying powers with the same base, add exponents: $a^m \times a^n = a^{m+n}$. 3. **Rewrite the equation:** $$5p^{-3} = 8 \times 5^{-2}$$ 4. **Express $p^{-3}$ as $\frac{1}{p^3}$:** $$5 \times \frac{1}{p^3} = 8 \times 5^{-2}$$ 5. **Simplify $5^{-2}$:** $$5^{-2} = \frac{1}{5^2} = \frac{1}{25}$$ 6. **Substitute back:** $$\frac{5}{p^3} = 8 \times \frac{1}{25} = \frac{8}{25}$$ 7. **Cross multiply to solve for $p^3$:** $$5 \times 25 = 8 \times p^3$$ $$125 = 8p^3$$ 8. **Divide both sides by 8:** $$p^3 = \frac{125}{8}$$ 9. **Take cube root of both sides:** $$p = \sqrt[3]{\frac{125}{8}} = \frac{\sqrt[3]{125}}{\sqrt[3]{8}} = \frac{5}{2}$$ **Final answer:** $p = \frac{5}{2}$ This corresponds to option E.