Dividing both sides by 2^x, we have:

1 + 2^(y - x) = 2^(30 - x)

We see that the right hand side is an integer power of 2. But the left hand side is 1 more than an integer power of 2, which can happen only if 2^(y - x) is also equal to 1. (When that happens, then the left side becomes 1 + 1 = 2, which is an integer power of 2.) (Also note that the only way in which 2^(y - x) can equal 1 is if (x - y) = 0.) Thus, the two sides of the equation can’t be equal unless y - x = 0, so that the left hand side is also an integer power of 2. So we have

1 + 2^0 = 2^(30 - x)

1 + 1 = 2^(30 - x)

2^1 = 2^(30 - x)

1 = 30 - x

x = 29

Since y - x = 0, so y = 29. Thus x + y = 29 + 29 = 58.

Answer: D
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2^x + 2^y = 2^30
(2^x+2^y)/2 = 2^29 (divide both sides by 2)
method 1: --> 2^29 is the average of 2^x and 2^y --> it is possible that 2^x = 2^y = 2^29 --> x = y = 29 --> x+ y = 29 + 29 = 58 (E)
method 2: --> 2^x + 2^y = 2.2^29 --> 2^x = 2^y = 2^29 --> x = y = 29 --> x + y = 29 + 29 = 58 (E)
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Can be solved by seeing a pattern in exponents
2^1 = 2^0 + 2^0
2^2 = 2^1 + 2^1
2^3 = 2^2 + 2^2
.
.
.
2^30 = 2^29 + 2^29

x + y = 58