To simplify the expression \( 3^{\sqrt{24}c^{16}} \), we can start by simplifying \( \sqrt{24} \).
First, factor \( 24 \): \[ 24 = 4 \times 6 = 4 \times 2 \times 3 \]
Now, we find the square root: \[ \sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} = 2\sqrt{6} \]
Next, substitute this back into the original expression: \[ 3^{\sqrt{24}c^{16}} = 3^{2\sqrt{6}c^{16}} \]
This is the simplified form of the expression. Thus, the simplified expression is: \[ 3^{2\sqrt{6}c^{16}} \]