1. **State the problem:** Simplify the expression $$\left(\frac{4p^{5}}{p^{-3}}\right)^{-2}$$ and write the answer using only positive exponents. 2. **Recall the rules:** - When dividing powers with the same base, subtract exponents: $$a^{m} \div a^{n} = a^{m-n}$$ - When raising a power to another power, multiply exponents: $$(a^{m})^{n} = a^{mn}$$ - Negative exponents mean reciprocal: $$a^{-m} = \frac{1}{a^{m}}$$ 3. **Simplify inside the parentheses:** $$\frac{4p^{5}}{p^{-3}} = 4p^{5 - (-3)} = 4p^{5 + 3} = 4p^{8}$$ 4. **Apply the outer exponent:** $$\left(4p^{8}\right)^{-2} = 4^{-2} \cdot p^{8 \times (-2)} = 4^{-2} p^{-16}$$ 5. **Rewrite with positive exponents:** $$4^{-2} = \frac{1}{4^{2}} = \frac{1}{16}$$ $$p^{-16} = \frac{1}{p^{16}}$$ 6. **Combine the results:** $$\frac{1}{16} \cdot \frac{1}{p^{16}} = \frac{1}{16p^{16}}$$ **Final answer:** $$\boxed{\frac{1}{16p^{16}}}$$