Simplify the expression. \[ \left(5 c^{4} d\right)^{2}\left(3 c d^{3}\right)^{2}=\square \]

To simplify the expression \(\left(5 c^{4} d\right)^{2}\left(3 c d^{3}\right)^{2}\), we first apply the power of a product rule, which tells us that \((ab)^n = a^n b^n\). 1. Starting with \(\left(5 c^{4} d\right)^{2}\): \[ \left(5 c^{4} d\right)^{2} = 5^{2} (c^{4})^{2} (d)^{2} = 25 c^{8} d^{2} \] 2. Next, for \(\left(3 c d^{3}\right)^{2}\): \[ \left(3 c d^{3}\right)^{2} = 3^{2} (c)^{2} (d^{3})^{2} = 9 c^{2} d^{6} \] Now combine the two results: \[ 25 c^{8} d^{2} \cdot 9 c^{2} d^{6} \] Multiply the coefficients and add the powers of like terms: \[ 25 \cdot 9 = 225 \] \[ c^{8} \cdot c^{2} = c^{8+2} = c^{10} \] \[ d^{2} \cdot d^{6} = d^{2+6} = d^{8} \] Putting it all together, we have: \[ 225 c^{10} d^{8} \] Thus, the simplified expression is: \[ \boxed{225 c^{10} d^{8}} \]