Okay, let's see. I need to find the square root of f(f(f(f(1)))), where f(x) is 2 raised to the power of x. Hmm, that sounds like applying the function f four times starting with 1, and then taking the square root of the result. Let me break this down step by step.

First, let's recall what the function f(x) = 2^x does. For any input x, it raises 2 to the power of x. So, applying f once to 1 would be 2^1, which is 2. Then applying f again to that result would be 2^2, which is 4. Wait, but the problem is asking for f(f(f(f(1)))), which is four applications of f. Then finally, take the square root of that number. Let me make sure I do each step correctly.

Starting with the innermost function: f(1). Since f(x) = 2^x, so f(1) = 2^1 = 2. Then, the next application is f(f(1)) = f(2). Again, f(2) = 2^2 = 4. Then f(f(f(1))) = f(4) = 2^4 = 16. Next, f(f(f(f(1)))) = f(16) = 2^16. Now, 2^16 is a large number. Let me calculate that. 2^10 is 1024, so 2^16 is 2^10 * 2^6 = 1024 * 64. Let's compute that: 1024 * 60 is 61,440, and 1024 * 4 is 4,096. Adding those together gives 61,440 + 4,096 = 65,536. So f(f(f(f(1)))) is 65,536.

Now, the problem asks for the square root of that result. So sqrt(65,536). Let me think about what number squared gives 65,536. I know that 256 squared is 65,536 because 256 * 256 is (2^8)^2 = 2^16, which is 65,536. Therefore, the square root of 65,536 is 256.

Wait, let me verify that. 256 * 256. Let's compute 250*250 = 62,500. Then, 250*6 = 1,500 and 6*250 = 1,500, and 6*6=36. Wait, no, that's not the right way. Alternatively, 256 * 256: break it down as (200 + 50 + 6) * (200 + 50 + 6). But maybe a simpler way is to note that 256 is 2^8, so 256 squared is (2^8)^2 = 2^16 = 65,536. Yes, that's correct. So the square root is indeed 256.

Therefore, the final answer should be 256. Let me recap to make sure I didn't make any mistakes. Starting with 1, applying f four times gives 2^1 = 2, then 2^2 = 4, then 2^4 = 16, then 2^16 = 65,536. The square root of 65,536 is 256. All steps seem correct. I think that's the right answer.
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