Simplify. \[ \left(\frac{y^{4}}{2 z^{3}}\right)^{4} \]

To simplify the expression \(\left(\frac{y^{4}}{2 z^{3}}\right)^{4}\), we apply the power of a quotient rule, which states that \(\left(\frac{a}{b}\right)^{n} = \frac{a^{n}}{b^{n}}\). Starting with the expression: \[ \left(\frac{y^{4}}{2 z^{3}}\right)^{4} = \frac{(y^{4})^{4}}{(2 z^{3})^{4}} \] Next, we simplify the numerator and the denominator separately. For the numerator: \[ (y^{4})^{4} = y^{16} \] For the denominator: \[ (2 z^{3})^{4} = 2^{4} (z^{3})^{4} = 16 z^{12} \] Combining these results gives us: \[ \frac{y^{16}}{16 z^{12}} \] Thus, the simplified expression is: \[ \frac{y^{16}}{16 z^{12}} \]