<Question>
What is the difference between the positive square root of 64 and the cube root of 64?
</Question>

<think>
<Plan_1>
Okay, let's see. The question is asking for the difference between the positive square root of 64 and the cube root of 64. Hmm, I need to figure this out step by step. First, I should probably find each root separately and then subtract them.

Starting with the square root part. The positive square root of 64. Wait, square root means a number that, when multiplied by itself, gives 64. So, what number times itself is 64? Let me think.

<prompt_1.1> Calculate the positive square root of 64. </prompt_1.1>

Now, moving on to the cube root of 64. Cube root means a number that, when multiplied by itself three times, gives 64. So, 4 times 4 is 16, 16 times 4 is 64.
So that's 4 as well. Wait, so both the square root and cube root of 64 are 4? That seems a bit coincidental, but mathem sometimes has those interesting coincidences.

But the question specifically mentions the positive square root.
So even though there's a negative square root (-8), we only take the positive one, which is 8. And the cube root, well, cube roots of positive numbers are positive, so it's 4.
-----

Is there a chance I mixed up the roots? Like, maybe the question meant the square root of the cube root or something?

<prompt_1.2> Consider if there was a misunderstanding of the roots and verify the interpretation of the question. </prompt_1.2>
</Plan_1>

<Execution_1>
<execution_1.1>
Let's list the squares: 1 squared is 1, 2 is 4, 3 is 9, 4 is 16, 5 is 25, 6 is 36, 7 is 49, 8 is 64. Oh, right! 8 times 8 is 64. So the positive square root of 64 is 8.
</execution_1.1>

<execution_1.2>
But no, the question clearly states the positive square root of 64 and the cube root of 64. So we're dealing with sqrt(64) and cbrt(64). So 8 and 4. Difference is 4.
</execution_1.2>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Wait, but hold on, the cube root.
The cube root of 64. Let me confirm. 4 cubed is 4*4=16, 16*4=64. Yep, so cube root of 64 is 4. So now, the problem asks for the difference between these two. So positive square root is 8, cube root is 4. So 8 minus 4?
That would be 4. So the difference is 4.

Wait, but is it 8 minus 4 or 4 minus 8? The question says "the difference between the positive square root of 64 and the cube root of 64".

<prompt_2.1> Determine the correct order for calculating the difference between the square root and cube root. </prompt_2.1>
-----

Maybe I should check with another method. For the square root, 64 is 2^6, so sqrt(2^6) is 2^(6/2) = 2^3 = 8.

<prompt_2.2> Verify the calculations using exponentiation and confirm the difference. </prompt_2.2>
</Plan_2>

<Execution_2>
<execution_2.1>
Depending on which one is subtracted from which. It just says the difference, so order might matter. But usually, when not specified, it's the first mentioned quantity minus the second. So positive square root (8) minus cube root (4) is 4. But maybe they want the absolute difference? The problem doesn't specify, but since both roots are positive, and 8 is larger than 4, the difference is 4 either way. Wait, 8 - 4 is 4, and 4 - 8 is -4, but difference is typically a positive value. Hmm, maybe I need to take the absolute value here.
</execution_2.1>

<execution_2.2>
For cube root, 64 is 2^6, so cube root is 2^(6/3) = 2^2 = 4. So that's another way to see it. So 8 - 4 = 4.
</execution_2.2>
</Execution_2>

<Plan_3>
Based on execution_2.1:
But the question says "the difference between the positive square root of 64 and the cube root of 64". Since it's just asking for the difference, not the absolute difference, but mathematically, difference can be signed.
However, in most contexts, especially in problems like this, they probably want the absolute value. Let me check.

If I subtract cube root from square root: 8 - 4 = 4. If I subtract square root from cube root: 4 - 8 = -4. But since difference is asked, and unless specified otherwise, it's possible they just want the numerical difference, which is 4. But maybe I should confirm.

Wait, let's see.

<prompt_3.1> Confirm the interpretation of the difference and consider both orders of subtraction. </prompt_3.1>
-----

Another way to check: if someone asks for the difference between A and B, it's usually |A - B| unless context says otherwise.

<prompt_3.2> Re-evaluate the interpretation of "difference" and confirm the order of subtraction. </prompt_3.2>

Based on execution_2.2, execution_1.2:
But to be thorough, let me check if there's any other interpretation.

<prompt_3.3> Verify the interpretation of the problem statement and confirm the solution. </prompt_3.3>
</Plan_3>

<Execution_3>
<execution_3.1>
For example, if I say, "What's the difference between 10 and 5?", it's 5. So it's the absolute value. So in this case, the answer is 4. Therefore, the difference is 4.
</execution_3.1>

<execution_3.2>
But in some cases, especially in algebra, it can be A - B. However, given that both numbers are positive and the square root is larger, the difference is 4. But to be safe, maybe the answer is 4. However, in many math problems, unless specified, difference is the absolute value. So 4.
</execution_3.2>

<execution_3.3>
The problem states: "the difference between the positive square root of 64 and the cube root of 64". So the two numbers in question are 8 and 4. So the difference is 8 - 4 = 4. Since the question doesn't specify absolute difference, but in common usage, difference is often considered positive. So 4 is the answer. I think that's it. I can't see any mistake here.
</execution_3.3>
</Execution_3>

<Final_answer>
Therefore, the difference between the positive square root of 64 and the cube root of 64 is $\boxed{4}$.
</Final_answer>
</think>