1. **State the problem:** Simplify the expression $$\frac{4v^{-9}}{2(v^2)^5}$$ so that it contains only positive exponents. 2. **Recall the exponent rules:** - Power of a power: $$(a^m)^n = a^{mn}$$ - Division of same bases: $$\frac{a^m}{a^n} = a^{m-n}$$ - Negative exponent rule: $$a^{-m} = \frac{1}{a^m}$$ 3. **Simplify the denominator:** $$(v^2)^5 = v^{2 \times 5} = v^{10}$$ 4. **Rewrite the expression:** $$\frac{4v^{-9}}{2v^{10}}$$ 5. **Divide the coefficients:** $$\frac{4}{2} = 2$$ 6. **Apply the division rule for exponents:** $$v^{-9} \div v^{10} = v^{-9 - 10} = v^{-19}$$ 7. **Combine the results:** $$2v^{-19}$$ 8. **Rewrite with positive exponents:** $$2v^{-19} = \frac{2}{v^{19}}$$ **Final answer:** $$\boxed{\frac{2}{v^{19}}}$$