Question: User: ing express
$\frac{(3x^{-2}y^{3})^{2}}{12x^{4}y^{-1}}$
1. **State the problem:** Simplify the expression $$\frac{(3x^{-2}y^{3})^{2}}{12x^{4}y^{-1}}$$.
2. **Apply the power to each factor inside the parentheses:**
$$(3x^{-2}y^{3})^{2} = 3^{2} \cdot (x^{-2})^{2} \cdot (y^{3})^{2} = 9x^{-4}y^{6}$$
3. **Rewrite the expression:**
$$\frac{9x^{-4}y^{6}}{12x^{4}y^{-1}}$$
4. **Combine like bases by subtracting exponents:**
$$= \frac{9}{12} \cdot x^{-4 - 4} \cdot y^{6 - (-1)} = \frac{9}{12} x^{-8} y^{7}$$
5. **Simplify the fraction:**
$$\frac{9}{12} = \frac{\cancel{3} \times 3}{\cancel{3} \times 4} = \frac{3}{4}$$
6. **Final simplified expression:**
$$\frac{3}{4} x^{-8} y^{7} = \frac{3 y^{7}}{4 x^{8}}$$
**Answer:** $$\frac{3 y^{7}}{4 x^{8}}$$