Make a Perfect Square

${2} ^ {124} + {2} ^ {126} +{ 2}^{n}$

= ${2}^{124}$ ( $1+4+{2}^{x}$ ). (Let n-124 be x)

We can see that ${2}^{124}$ is a square number. So 1 + 4 + ${2}^{x}$ should also be square number.

So the least value of ${2}^{x }$ = 4

Hence x=2 and x=n-124 Therefore n=126

Since the said equation is a perfect square

by binomial expansion

Then we can simply equate that

$(a+b)^2 = a^2 + b^2 + 2ab$ - $i$

$2^{124} = (2^{62})^{2}$

$2^{126} = (2)(2^62)(2^{\dfrac{n}{2}})$

$2^{n } = (2^{\dfrac{n}{2}})^2$ - $ii$

Hence by comparing $ii$ by $i$ , we get $n = 126$