First of all, your answer is obviously wrong. When x=y=z, A x * b y * c z = 1 cannot be deduced. You can only infer that the first three are equal. Go back and look at the topic carefully first.

Here's a hint.

Suppose your theme is

a^x * b^y * c^z=a^y * b^z *c^x = a^z * b^x * c^y=k

Take the logarithm of the above formula (no matter what the base is) and you get

x*lg(a)+y*lg(b)+z*lg(c)=lg(k)

x*lg(b)+y*lg(c)+z*lg(a)=lg(k)

x*lg(c)+y*lg(a)+z*lg(b)=lg(k)

Solve the above equation

x=y=z=lg(k)/(lg(a)+lg(b)+lg(c))