To determine the exponent that would make the two expressions equivalent, we start with the expressions you've provided:
- \((3)^3 \cdot \pi\)
- \(3^{24}\)
We want to find the value of \(x\) such that:
\[ (3^3 \cdot \pi)^x = 3^{24} \]
First, let's simplify the expression on the left side:
\[ (3^3 \cdot \pi)^x = (3^3)^x \cdot \pi^x = 3^{3x} \cdot \pi^x \]
Now, we set this equal to \(3^{24}\):
\[ 3^{3x} \cdot \pi^x = 3^{24} \]
For equality to hold, the powers of 3 on both sides must match, and since \(\pi^x\) does not appear on the right side, \(x\) must equal 0 for that term to vanish. Therefore:
\[ 3x = 24 \implies x = \frac{24}{3} = 8 \] And since \(\pi^x\) must equal \(1\), we have the term \(x = 0\).
However, if we are only considering the powers of \(3\), then \(x\) as \(8\) is sufficient to equate the powers of \(3\) in both expressions.
So, the answer to your inquiry is that the exponent \(x\) which makes the two expressions equivalent (ignoring the \(\pi\) component) is \(x = 8\).
If you meant the final result to relate singularly to \(2715\) or \(21\), I don’t see a direct correlation to that number based on your question. Please clarify if those numbers have a specific relation you wanted to explore!
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