We are asked to simplify the expression $(\frac{3xy^{-3}}{z^3})^4$ and express the answer with positive exponents.

1. Problem Description

We are asked to simplify the expression $(\frac{3xy^{-3}}{z^3})^4$ and express the answer with positive exponents.

2. Solution Steps

First, apply the power of a quotient rule: $(\frac{a}{b})^n = \frac{a^n}{b^n}$.

$(\frac{3xy^{-3}}{z^3})^4 = \frac{(3xy^{-3})^4}{(z^3)^4}$

Next, apply the power of a product rule: $(ab)^n = a^n b^n$.

$\frac{(3xy^{-3})^4}{(z^3)^4} = \frac{3^4 x^4 (y^{-3})^4}{(z^3)^4}$

Now, apply the power of a power rule: $(a^m)^n = a^{mn}$.

$\frac{3^4 x^4 y^{-12}}{z^{12}}$

Since $3^4 = 81$, we have

$\frac{81 x^4 y^{-12}}{z^{12}}$

We need to express the answer with positive exponents. Recall that $a^{-n} = \frac{1}{a^n}$.

So, $y^{-12} = \frac{1}{y^{12}}$.

Therefore,

$\frac{81 x^4 y^{-12}}{z^{12}} = \frac{81 x^4}{y^{12} z^{12}}$

3. Final Answer

$\frac{81x^4}{y^{12}z^{12}}$