Find all the infinite cardinals $a$ such that $a^4 + 16^a = 16^a$

Question

Find all the infinite cardinals $a$ such that $a^4 + 16^a = 16^a$.

I think this is equivalent to $a^4 + (2^a)^4 = (2^a)^4$, but I don't know how to prove the latter.

Any help would be appreciated.

Answer 1

For all infinite cardinals under AC, $a^4=a, 16^a=2^a \gt a, a+b=\max(a,b)$