Prime factorization, Exponents

[turdferguson]

This was taken from a math contest a few months ago.

Homework Statement

x^x*y^y=z^z

find z if:
x=2⁸ * 3⁸
y=2¹² * 3⁶

Homework Equations

there's undoubtably some trick, but I have yet to find it

The Attempt at a Solution

Dont even think about calculator

I showed my math teacher, and he was able to find that z^z=2^(211*37*11) * 3^(211*37*7)

or 2^49268736 * 3^31352832

How do you get z alone?

 

[Werg22]

$$z^{z} = 2^{2^{11}*3^{7}*11} * 3^{2^{11}*3^{7}*7} = ({2^{11}})^{2^{11}*3^{7}}*({3^{7}})^{2^{11}*3^{7}} = ({2^{11}*3^{7}})^{2^{11}*3^{7}}$$

$$z = {2^{11}*3^{7}}$$