TITLE: Powers containing every digit equally often
QUESTION [5 upvotes]: There are several nontrivial powers containing every digit equally often, for example
$32043^2$ $2158479^3$ $69636^4$ $643905^5$ $3470187^6$
A necessary condition for a power with the desired property is that the base is
 divisible by $3$ because the power must be a multiple of $9$.
My questions :

Are there powers of $3$,$6$ and $9$ with the desired property ?
Are there infinite many nontrivial powers with the desired property ?
I did not find a $10th$ power with the desired property yet. Is there one ?

REPLY [2 votes]: Here is a $\frac{1}{3}$th of an answer.
There is an 10th power with the property. Here is a list of the smallest solutions for a given power:
Edit to keep the post small, see Jeppes comment for a list up to 24:

10:  81785058