Okay, so I need to find the difference between the positive square root of 64 and the cube root of 64. Let me start by breaking down each part step by step. First, understanding what a square root and a cube root are. 

The square root of a number is a value that, when multiplied by itself, gives the original number. Since the question mentions the "positive" square root, that means I don't have to consider the negative one. For 64, the square root would be the number that times itself equals 64. Let me recall, 8 times 8 is 64. So the square root of 64 is 8. That part seems straightforward.

Now, the cube root of 64. A cube root of a number is a value that, when multiplied by itself three times, gives the original number. Again, the question specifies the positive cube root, so I don't need to worry about negative numbers here. Let's think, what number multiplied by itself three times is 64? Let's start with 4: 4 times 4 is 16, and 16 times 4 is 64. So 4 cubed is 64. Therefore, the cube root of 64 is 4.

Now, the problem asks for the difference between these two roots. So I need to subtract the smaller number from the larger one. The square root is 8 and the cube root is 4. So 8 minus 4 equals 4. Therefore, the difference should be 4. 

Wait, let me double-check my calculations to make sure I didn't make a mistake. For the square root: 8*8=64, correct. Cube root: 4*4*4=64, yes. Then 8-4=4. Hmm, seems right. 

Alternatively, maybe I should verify if there's another way to approach this. For example, could there be another number whose cube is 64? Let's see. Let's try 3: 3*3=9, 9*3=27. No, that's 27. 5: 5*5=25, 25*5=125. That's too big. So 4 is indeed the cube root. 

Another check: 64 divided by 4 is 16, then divided by 4 again is 4. So 4*4*4=64. Yep. 

So square root is 8, cube root is 4, difference is 4. Seems straightforward. I don't think there's any trick here, like considering multiple roots or something. The question specifies the positive roots, so negatives are out of the question. 

Therefore, the final answer should be 4. Let me just make sure I didn't misread the question. The difference between the positive square root and the cube root. Yes, so 8 - 4 = 4. Correct. 

I think that's all. No steps missing here. The calculations are simple, but it's easy to mix up the roots. For example, confusing the cube root with the square root, but since the cube root of 64 is 4 and the square root is 8, their difference is indeed 4. 

Another way to look at it: if I write both roots in exponential form. The square root is 64^(1/2) which is 8, and the cube root is 64^(1/3) which is 4. So subtracting them gives 8 - 4 = 4. Same result. 

Therefore, I'm confident the answer is 4.
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