15. \( \left(2 x^{3} y^{2} z^{2}\right)^{3}\left(x^{2} z\right)^{4} \)

To find the expression \( \left(2 x^{3} y^{2} z^{2}\right)^{3}\left(x^{2} z\right)^{4} \), we first apply the laws of exponents. Starting with \( \left(2 x^{3} y^{2} z^{2}\right)^{3} \), we distribute the exponent of 3: \[ = 2^{3} (x^{3})^{3} (y^{2})^{3} (z^{2})^{3} = 8 x^{9} y^{6} z^{6} \] Next, we simplify \( \left(x^{2} z\right)^{4} \): \[ = (x^{2})^{4} (z)^{4} = x^{8} z^{4} \] Now, we can combine the two parts: \[ 8 x^{9} y^{6} z^{6} \cdot x^{8} z^{4} \] Applying the laws of exponents for \( x \) and \( z \): \[ = 8 x^{9 + 8} y^{6} z^{6 + 4} = 8 x^{17} y^{6} z^{10} \] Thus, the final expanded expression is: \[ 8 x^{17} y^{6} z^{10} \]