To simplify \((xy^3z^4)^4\), you can apply the power of a product rule, which states that \((a \cdot b)^n = a^n \cdot b^n\).
In this case, you can apply the exponent to each factor inside the parentheses:
\[ (xy^3z^4)^4 = x^4 \cdot (y^3)^4 \cdot (z^4)^4 \]
Next, simplify each term using the power of a power rule, which states that \((a^m)^n = a^{m \cdot n}\):
\[ (y^3)^4 = y^{3 \cdot 4} = y^{12} \] \[ (z^4)^4 = z^{4 \cdot 4} = z^{16} \]
Now combine everything:
\[ x^4 \cdot y^{12} \cdot z^{16} \]
Thus, the simplified expression is:
\[ \boxed{x^4 y^{12} z^{16}} \]