Just looking at this we can see that
Which would make z = -1.
You could also make the observation that 1/6+1/3=1/2.
More observation than guess and check.
You could play around and see what you can whip up.
Divide by :
Now, since 1+2=3, z must equal -1.
Just playing around, you can always try other things.
Look into the Lambert W function for tricky exponent problems.