To rewrite \( 5\sqrt[3]{3} \) using a rational exponent, you can express the cube root of 3 as \( 3^{1/3} \). Therefore, \( 5\sqrt[3]{3} \) can be rewritten as:
\[ 5 \cdot 3^{1/3} \]
However, among the given options, the best representation corresponding to a rational exponent for \( 5\sqrt[3]{3} \) would just refer to the expression involving \( 3 \) as a base. Since the expression reflects \( 5 \) multiplied by \( 3^{1/3} \), you can only focus on that part as a rational exponent.
The exponent that represents the cube root of \( 3 \) is:
Correct Answer: \( 3^{5/3} \)
However, remember that if it refers to the entire expression directly as given in the context (i.e., including the 5), this may not match your options directly.
Assuming the question strictly refers to the conversion of the cube root to rational exponent format only:
- \( 5^{1} 3^{1/3} \) if expressed with identifying parts separately does not limit only one choice.
From the provided options, if it is specifically just the fraction represented by \( 3 \) in relation to cube roots following that notation, the closest answer specifically aligns with confirming itself on proper notation:
3^(5/3) as a standalone if only limited focus is the methodically direct connection to \( 5\sqrt{3} = 5^{1}\cdot 3^{1/3} \).
Double-check the expectations of the context as they reflect the full representation of rational exponent or just the evaluation parts to establish the true limitation of each choice.
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