To express \( 312m \) with a rational exponent, we want to write it in the form of \( (a \cdot b)^c \). The options you provided seem to suggest different ways to express this.
The two expressions you listed that maintain the integrity of both parts (the number and the variable) in a single rational exponent are:
- \( (312m)^{1/6} \)
- \( 312^{1/6}m^{1/6} \)
However, if we are strictly looking for a single expression with a rational exponent, the most appropriate form would be:
Option 2: \( (312m)^{1/6} \)
This indicates that both \( 312 \) and \( m \) are raised to the power of \( \frac{1}{6} \).
So, it's best to choose \( (312m)^{1/6} \) as the answer.