To simplify the expression \(32 \cdot 3^{-5}\), we start by converting \(32\) into a power of \(3\) if possible, but it actually isn't a power of \(3\). Instead, we will analyze the expression as is.
Recognizing that \(3^{-5}\) can be rewritten, we have:
\[ 32 \cdot 3^{-5} = \frac{32}{3^5} \]
Now let's calculate \(3^5\):
\[ 3^5 = 243 \]
Therefore, substituting back, we have:
\[ 32 \cdot 3^{-5} = \frac{32}{243} \]
Next, the choices provided do not include this expression directly, but we need to express this as one of the given answer choices.
None of the options match, but if we consider the powers of \(3\) within the choices:
- The options provided are all in terms of negative exponents of \(3\).
Now, we analyze if any option can be expressed similarly.
Reviewing the options provided:
- \(\frac{1}{3^{-7}} = 3^{7}\) (not equivalent)
- \(\frac{1}{3^{-3}} = 3^{3}\) (not equivalent)
- \(\frac{1}{3^{3}} = 3^{-3}\) (not equivalent)
- \(\frac{1}{3^{7}} = 3^{-7}\) (not equivalent)
It appears there are no direct matches. However, in summary:
The expression \(32 \cdot 3^{-5}\) can be simplified to \(\frac{32}{243}\), which does not precisely correlate with any presented options. If we assume \(32\) could also lead to a different form within the relation to \(3^{-x}\), we observe a mismatch in representation.
Thus, unless further clarification is provided, my suggestion stands as \(\frac{32}{243}\) being the correct simplification. If you are required to select from these options, please double-check for possible errors, as none indeed represent the transformation of \(32 \cdot 3^{-5}\) accurately.