Question: Simplify.
$(-3y^{-3}z^{4})^{2}$
Write your answer using only positive exponents.
1. **State the problem:** Simplify the expression $$(-3y^{-3}z^{4})^{2}$$ and write the answer using only positive exponents.
2. **Recall the power of a product rule:** $$(ab)^{n} = a^{n}b^{n}$$ and the power of a power rule: $$(a^{m})^{n} = a^{mn}$$.
3. **Apply the power to each factor:**
$$(-3)^{2} (y^{-3})^{2} (z^{4})^{2}$$
4. **Calculate each term:**
$$(-3)^{2} = 9$$
$$(y^{-3})^{2} = y^{-6}$$
$$(z^{4})^{2} = z^{8}$$
5. **Combine the results:**
$$9 y^{-6} z^{8}$$
6. **Rewrite with positive exponents:**
Since $y^{-6} = \frac{1}{y^{6}}$, the expression becomes
$$\frac{9 z^{8}}{y^{6}}$$
**Final answer:**
$$\boxed{\frac{9 z^{8}}{y^{6}}}$$