To use the product rule to rewrite \(8^t + 8^v\), we first notice that \(8\) can be expressed as \(2^3\). Then, we can rewrite \(8^t\) and \(8^v\):
\[ 8^t = (2^3)^t = 2^{3t} \] \[ 8^v = (2^3)^v = 2^{3v} \]
We can factor out the common base, but the expression \(8^t + 8^v\) cannot be simplified directly to a product without additional context.
However, if we consider the sum of two exponentials of the same base, we can rewrite the expression in terms of their products. Notice the \(8^t + 8^v\) can also be expressed as:
\[ 8^t + 8^v = 8^t(1 + 8^{v-t}) \quad \text{(Factoring out } 8^t\text{)} \]
This doesn't directly correspond to any of the given options.
If we are looking to match one of the choices given, \(64^t+v\) and others may not be directly derived from \(8^t + 8^v\) using simple product rules, as that represents an actual sum rather than a product.
Here if we consider \(8^{t+v}\):
\[ 8^{t+v} can be represented as 64^{(t+v)/3} \text{ (as } 64 = 8^2) \]
But that doesn't match the options either.
None of the choices directly correspond to the accurate representation of \(8^t + 8^v\). The closest representation from those provided may be mistakenly offered, but:
- Final conclusion: None of the provided options correctly represents \(8^t + 8^v\) using the product rule appropriately without additional information or context, as product rule typically applies to multiplication, not addition.