When aligning 2 equations leaves gaps in shorter equation, can I center the shorter bits?

Questions
,
1

It’s a lot easier to show than to describe!
I aligned my equation like this:

$
  cases(
    2^2 & =4 <=> 4^(1/2) & =2=sqrt(4),
    2 dot 2 & =4 <=> 4/2 & =2=4 dot 1/2
  )
$

Although for the extra stylistic touch, I though it would be nice to place the 2^2, the 4/2 and the sqrt(4) in the middle, rather than to the left. What would you suggest?