To rewrite \( \sqrt[3]{82} \) using exponents, we can express it as \( 82^{1/3} \).
Now, we can rewrite the expression:
\[ \sqrt[3]{82} = 82^{1/3} \]
Therefore, the rational number that could be used as an exponent is \( \frac{1}{3} \).
Next, let's clarify the expression given:
If the expression you provided is \( \sqrt[3]{82} \), we have already expressed it as \( 82^{1/3} \).
If we are to simplify further, we note that \( \sqrt[3]{82} \) does not simplify into a simpler rational number since 82 does not have any perfect cube factors. Hence, the fully simplified expression is:
\[ \sqrt[3]{82} = 82^{1/3} \]
So, summarizing:
The rational number that could be used as an exponent is \( \frac{1}{3} \).
And \( 82^{1/3} \) remains as is because it cannot be simplified further.