The language L = {a^nb^m : n =2^m} can be shown to not be context-free by using the pumping lemma for context-free languages.

The pumping lemma states that if a language L is context-free, then there exists a number p (the pumping length) such that any string s in L of length greater than or equal to p can be divided into three substrings x, y, and z such that:

|xy| <= p

|y| > 0

for all i >= 0, xy^i z is in L

Let's take the string "a^4b^2" which is in the language L.

We can divide the string into x = "a^2", y = "a^2", z = "b^2"

Now we can pump y any number of times to get "a^4b^2, a^6b^2, a^8b^2,..."

All of these new strings generated by pumping y are not in the language L because n is not equal to 2^m, which breaks the rule of the language L.

Therefore, we can conclude that the language L = {a^nb^m : n =2^m } is not context-free.