Two Nasty Exponent Mistakes

Nasty Mistake 1: $\sqrt{(x+y)^2} \neq x+y$
Remember with a square root, the result must be non-negative. This should actually be $|x+y|$.

Nasty Mistake 2: $2^{2^{2^{2}}} \neq 2^8$
If there are no parenthesis, you need to evaluate exponents from the top-right, working downwards:

$2^{2^{(2^{2})}} = 2^{(2^{4})} = 2^{16} = 65536$