1. **State the problem:** Simplify the expression $$\frac{2x^{-3}y^{4}}{4x^{-1}y^{-2}}$$ and write it using only positive indices. 2. **Recall the rules:** - When dividing powers with the same base, subtract the exponents: $$\frac{a^{m}}{a^{n}} = a^{m-n}$$. - Negative exponents mean reciprocal: $$a^{-m} = \frac{1}{a^{m}}$$. 3. **Apply the rules to the expression:** $$\frac{2x^{-3}y^{4}}{4x^{-1}y^{-2}} = \frac{2}{4} \times \frac{x^{-3}}{x^{-1}} \times \frac{y^{4}}{y^{-2}}$$ 4. **Simplify coefficients:** $$\frac{2}{4} = \frac{\cancel{2}}{2 \times \cancel{2}} = \frac{1}{2}$$ 5. **Simplify the x terms using exponent subtraction:** $$x^{-3 - (-1)} = x^{-3 + 1} = x^{-2}$$ 6. **Simplify the y terms using exponent subtraction:** $$y^{4 - (-2)} = y^{4 + 2} = y^{6}$$ 7. **Rewrite the expression:** $$\frac{1}{2} x^{-2} y^{6}$$ 8. **Convert negative exponent to positive:** $$x^{-2} = \frac{1}{x^{2}}$$ 9. **Final simplified expression:** $$\frac{y^{6}}{2x^{2}}$$