To express \( v^{216} \) with a rational exponent, we need to first understand that the expression can be represented using properties of exponents.
The expression \( v^{216} \) can be rewritten using rational exponents as:
\[ v^{216} = (v^{1})^{216} = v^{\frac{216}{1}} = v^{216} \]
However, since \( v^{216} \) or \( v^{n} \) can also be interpreted as the root of a number when expressed with a rational exponent, if we consider it as \( v^{\frac{216}{n}} \), we can say that any equivalent expression with a rational exponent would have a numerator of 216.
Among the potential choices for rational exponents (3, 1, -3), the focus would be on where 216 could be expressed as a product or a division of rational numbers.
None of the choices \( (3, 1, -3) \) appear to directly apply as they stand. However, if we were to think of \( v^{216} \) equivalently involving \( 216 = 72 \cdot 3 \) or any breakdown, the rational representation could take various forms contingent on common factors or roots, which none of the choices seem to represent verbatim.
It's possible that the intention was to identify a simpler representation, such as \( v^{3} \) that provides a clearer relationship through manipulation, albeit not strictly equal.
Therefore, if I were to simplify and choose based on the provided options:
- The rational exponent likely intended to represent the base, while the derivation itself seems emboldened mainly to elucidate the "exponent" as it progresses through overall values.
But, being cautious about direct equivalence, I would consider simplifying as \( 3 \) if it stems from cubic roots or comparative expressions leading to \( 216 \) under fractional exponentials.
Hence, if choosing based strictly on rational dividing or a cube, then 3 would be your representation related back progressively.
Thus, \( 3 \) is the rational exponent that might be interpreted in a simplified expression from the structure given.
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