Okay, let's see. I need to solve the equation 2^(2x) equals 256 to the power of 1/2. Hmm, first, maybe I should simplify both sides of the equation to make it easier to solve for x. Let me start by understanding what each side is.

On the left side, we have 2 raised to the power of 2x. That's straightforward. On the right side, 256^(1/2) is the square root of 256. Wait, 256 is a number I recognize. Let me confirm, 2^8 is 256, right? Because 2^10 is 1024, so 2^8 should be 256. So 256 is 2^8. Therefore, 256^(1/2) would be the square root of 2^8. The square root of 2^8 is 2^(8*(1/2)) which is 2^4. Because when you take a square root, you divide the exponent by 2. So 8 divided by 2 is 4. Therefore, 256^(1/2) is 2^4.

So now the equation becomes 2^(2x) = 2^4. Since the bases are the same (both are base 2), I can set the exponents equal to each other. That means 2x = 4. Then solving for x, I divide both sides by 2, so x = 4/2 = 2. That seems straightforward.

Wait, let me double-check. Let's plug x = 2 back into the original equation to verify. Left side: 2^(2*2) = 2^4 = 16. Right side: 256^(1/2) = sqrt(256) = 16. Yes, both sides equal 16. So x = 2 is indeed the correct solution.

Alternatively, maybe I can approach this using logarithms. Taking the natural logarithm of both sides: ln(2^(2x)) = ln(256^(1/2)). Using the logarithm power rule, that's 2x ln(2) = (1/2) ln(256). Then solving for x: x = [ (1/2) ln(256) ] / [2 ln(2) ] = (ln(256) ) / (4 ln(2)). Since 256 is 2^8, ln(256) is 8 ln(2). So substituting that in: x = (8 ln(2)) / (4 ln(2)) = 8/4 = 2. Same result. So that confirms it again.

Another way: Let's express 256 as 2^8. Then 256^(1/2) is (2^8)^(1/2) = 2^(8*(1/2)) = 2^4, which is 16. So the equation becomes 2^(2x) = 16. Then 16 is 2^4, so 2x = 4, so x = 2. All methods lead to the same answer, so I think that's solid.

I don't see any mistakes here. The key steps were recognizing that 256 is a power of 2, simplifying the square root, and then equating the exponents since the bases are the same. Alternatively, using logarithms also works, but it's more involved. The first method is more straightforward. Either way, the answer is x = 2.
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