1. **State the problem:** Simplify the expression $$\left(\frac{x^{-3} y^{3} z^{0}}{x^{4} y^{0} z^{-4} (y^{0})^{3}}\right)^{4}$$. 2. **Recall the rules:** - Any variable to the zero power is 1, i.e., $z^{0} = 1$ and $y^{0} = 1$. - When dividing powers with the same base, subtract exponents: $\frac{a^{m}}{a^{n}} = a^{m-n}$. - When raising a power to another power, multiply exponents: $(a^{m})^{n} = a^{mn}$. 3. **Simplify inside the parentheses first:** - Replace $z^{0} = 1$ and $(y^{0})^{3} = 1^{3} = 1$. - The expression inside becomes: $$\frac{x^{-3} y^{3} \cdot 1}{x^{4} \cdot 1 \cdot z^{-4} \cdot 1} = \frac{x^{-3} y^{3}}{x^{4} z^{-4}}$$ 4. **Apply division of powers:** $$x^{-3} / x^{4} = x^{-3-4} = x^{-7}$$ $$y^{3} / 1 = y^{3}$$ $$1 / z^{-4} = z^{4}$$ So the expression inside parentheses is: $$x^{-7} y^{3} z^{4}$$ 5. **Raise the entire expression to the 4th power:** $$(x^{-7} y^{3} z^{4})^{4} = x^{-7 \times 4} y^{3 \times 4} z^{4 \times 4} = x^{-28} y^{12} z^{16}$$ 6. **Final simplified expression:** $$\boxed{x^{-28} y^{12} z^{16}}$$ This means the expression simplifies to $x^{-28} y^{12} z^{16}$, where negative exponents indicate reciprocal powers if desired.