To simplify \( 3^{\sqrt{24}c^{16}} \), let's break it down step by step.
First, simplify \( \sqrt{24} \): \[ \sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} = 2\sqrt{6}. \]
Now substitute this back into the expression: \[ 3^{\sqrt{24}c^{16}} = 3^{2\sqrt{6}c^{16}}. \]
Next, let's break this down further:
- \( 3^{2\sqrt{6}c^{16}} \) can be expressed as \( (3^{\sqrt{6}})^{2c^{16}} \).
Since we have \( c^{16} \), we can factor it out as well: \[ = (3^{\sqrt{6}})^{2} \cdot 3^{c^{16}}. \]
However, it seems the provided options don't relate to the simplification directly, as it relates more to mathematical expressions than simplification of powers.
If we look at the provided options:
a) \( 2c^8\sqrt{6} \)
b) \( 2\sqrt{3}c^{16} \)
c) \( 6c^8\sqrt{6} \)
d) \( 2c^5\sqrt{3c} \)
None seem to match. Let's revert to the original expression assuming it might refer to different parts rather than an exponential function. Perhaps the options imply different manipulations or representations using separate values of \( c \).
If this was meant for \( 3^{\sqrt{24}}c^{16} \) expanding for numerical simplification:
Let’s try to regroup based on those:
- \( 4 = 2^2 \Rightarrow 3^{\sqrt{4}} \) gives proportional constants leading up to those options.
- A better representation might yield like terms.
In clearer terms, given that my breakdown has expanded significantly, could you clarify which components exactly need working or if \( \sqrt{c} \) terms should be involved as potential simplifying assumptions? Otherwise, with initial formulations made clear, we circled around options instead of precisely landing on target outcomes, so I’d adjust computations accordingly with narrowed focus.
Final output given the absence of clear equivalence: Unless more context shapes those workings toward a closer target output, none appear to match perfectly without further clarifications bestowed through component fitting.