1. **Problem:** Solve for $p$ in the equation $5p^{-3} = 8 \times 5^{-2}$. 2. **Formula and rules:** Recall that $a^{-n} = \frac{1}{a^n}$ and properties of exponents allow us to manipulate terms. 3. **Step-by-step solution:** - Start with the equation: $$5p^{-3} = 8 \times 5^{-2}$$ - Rewrite $5^{-2}$ as $\frac{1}{5^2} = \frac{1}{25}$: $$5p^{-3} = 8 \times \frac{1}{25} = \frac{8}{25}$$ - Divide both sides by 5: $$p^{-3} = \frac{8}{25} \times \frac{1}{5} = \frac{8}{125}$$ - Recall $p^{-3} = \frac{1}{p^3}$, so: $$\frac{1}{p^3} = \frac{8}{125}$$ - Taking reciprocals: $$p^3 = \frac{125}{8}$$ - Take cube root of both sides: $$p = \sqrt[3]{\frac{125}{8}} = \frac{\sqrt[3]{125}}{\sqrt[3]{8}} = \frac{5}{2}$$ 4. **Answer:** $p = \frac{5}{2}$ This corresponds to option E.