If 2^x + 2^y = 2²², what is the value of x + y?

Question: If 2^x + 2^y = 2²², what is the value of x + y?

Solution:
Here,,
2^x + 2^y=2²²
⇒ 2^x(1 + 2^{y - x})=2²²

2^x must be a power of 2 that divides 2²²
(1 + 2y - x) must be a power of 2
Only way (1 + 2y - x) is a power of 2 is if, y - x = 0

So, y - x = 0
⇒ y = x

∴ 2^x + 2^x = 2²²
⇒ 2.2^x = 2²²
⇒ 2^{1 + x}=2²²
⇒ 1 + x = 22
∴ x = 21
∴ x + y = 21 + 21 = 42