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Bot GPT 4-o mini To simplify the expression \( 216\sqrt{32713} \) using exponent properties, we start by analyzing the components individually.
First, observe that:
\[
216 = 6^3
\]
Next, we have the term involving the square root:
\[
\sqrt{32713} = 32713^{1/2}
\]
Combining these, we rewrite the entire expression:
\[
216\sqrt{32713} = 6^3 \cdot 32713^{1/2}
\]
Evaluating \( 6^3 \) gives us 216. However, upon evaluating \( 32713^{1/2} \), we keep it as is.
The problem requires us to simplify \( 6^3 \) and \( 32713^{1/2} \).
If you reference how many factors of 2 are in these expressions, we can consider the possible choices. To evaluate accurately, we should know the prime factorization and powers of a base of 2, which will help us in determining mutual equivalents.
However, without being given specific relationships between the bases of the resulting expressions (2, 8, 12, and 18), we'll presume that the original prompt is looking for a simple equivalent numerical value rather than a fractional or exponential representation.
Out of the options provided:
- 2
- 8
- 12
- \( \frac{1}{2} \)
- 18
Recall that \( 6^3 = 216 \), thus indicating that without much calculation, the answer cannot simplify to a fraction nor the number 2, as 6 is not divided.
Without observing a match for direct proportionality among these concrete selections, intuitively checking:
- \( 8 \) is \( 2^3 \),
- \( 12 \) is \( 3*4 \),
- while \( 18 \) is \( 2 \times 9 \).
Therefore, the answer should fundamentally revert back to a factorial play of \( 8 \) (since \( 8 = 2^3\)) in theoretical foundational numeric output concerning potential binary relations.
Given there may seem an omission of explicit detail, we consider if further context references deeper specifics revolving around the choice options analyzed.
**Thus the expected equivalent expressed in these broader terms would seem to be:**
**Choice:** 12 (which can be aligned with algebraic outcomes in exponents repeatedly).