jmc

Of the numbers from $1$ to $24$, eight of them are multiples of $3$, so that gives us an exponent of $8$.

Now, two of the numbers are multiples of $3^2=9$, so each of them has $3$ as a factor twice. We've already counted them one each, so we need to count them each one more time. This adds another $2$ to the exponent.

The next thing to check is whether any of the numbers have $3$ as a factor three times. Happily, $3^3=27>24$, so we don't have any of those.

Our total exponent is $8+2=\boxed{10}$.