Okay, let's try to solve this equation: the cube root of (x squared minus 4x plus 4) equals 16. Hmm, so the equation is ∛(x² - 4x + 4) = 16. Alright, first, I need to isolate the cube root and then get rid of the cube root by cubing both sides. Let me write that down step by step. 

First, let me recall that if ∛(A) = B, then A must be equal to B³. So applying that here, the left side is ∛(x² - 4x + 4), which equals 16. Therefore, if I cube both sides, I should get x² - 4x + 4 = 16³. Let me compute 16 cubed. 16 times 16 is 256, and then 256 times 16 is 4096. So 16³ is 4096. Therefore, the equation becomes x² - 4x + 4 = 4096. 

Now, I need to solve this quadratic equation. Let's subtract 4096 from both sides to set it to zero: x² - 4x + 4 - 4096 = 0. Simplifying that, 4 - 4096 is -4092. So the equation is x² - 4x - 4092 = 0. 

Now, solving the quadratic equation x² - 4x - 4092 = 0. Quadratic equations can be solved using the quadratic formula: x = [-b ± √(b² - 4ac)] / (2a). Here, a = 1, b = -4, c = -4092. Plugging those into the formula, we get x = [4 ± √(16 + 16368)] / 2. Wait, let's compute the discriminant first. 

Discriminant D = b² - 4ac = (-4)² - 4*1*(-4092) = 16 + 16368 = 16384. Hmm, 16384 is a perfect square because 128² is 16384. Let me check: 128*128. 120² is 14400, 8² is 64, and 2*120*8 is 1920. So 14400 + 1920 + 64 = 16384. Yes, so √16384 = 128. 

Therefore, the solutions are x = [4 ± 128]/2. Let's compute both possibilities. 

First solution: (4 + 128)/2 = 132/2 = 66. 

Second solution: (4 - 128)/2 = (-124)/2 = -62. 

But the problem asks for the positive value of x, so x = 66. 

Wait, let me verify this solution to make sure I didn't make any mistakes. Let's plug x = 66 back into the original equation. 

Compute x² - 4x + 4: 66² is 4356, 4x is 264. So 4356 - 264 + 4. 4356 - 264 is 4092, plus 4 is 4096. Then the cube root of 4096 is indeed 16, since 16³ is 4096. So that checks out. 

What about x = -62? Let's check that too. (-62)² is 3844, 4x is -248. So 3844 - (-248) + 4. That's 3844 + 248 + 4 = 3844 + 252 = 4096. Again, cube root of 4096 is 16. So both solutions satisfy the equation, but since we need the positive value, x = 66 is the answer. 

Wait, but let me think again. The original equation is ∛(x² -4x +4) =16. The cube root function is defined for all real numbers, so even if x² -4x +4 is negative, the cube root is still real. However, in this case, when we solved the equation, we found that x² -4x +4 equals 4096, which is positive. So both solutions for x result in the expression inside the cube root being 4096, which is positive. Therefore, both are valid. But since the problem asks for the positive value, we choose 66. 

Is there any possibility that I made a mistake in solving the quadratic equation? Let me check the discriminant again. The quadratic equation is x² -4x -4092 = 0. So a=1, b=-4, c=-4092. Discriminant is (-4)^2 -4*1*(-4092)=16 +16368=16384. Correct. Square root of 16384 is 128. So solutions are [4 ±128]/2. 4+128=132, divided by 2 is 66. 4-128=-124, divided by 2 is -62. Correct. 

Another way to check is to factor the quadratic. Let's see if x² -4x -4092 can be factored. Looking for two numbers that multiply to -4092 and add to -4. Hmm, 4092 is a big number. Let's see, 4092 divided by 4 is 1023. 1023 is 3*341. 341 is 11*31. So factors of 4092 would be ... maybe 62 and 66? Let's check: 62*66 is 4092. Wait, 62*66 is (60+2)*(60+6)=60² +60*6 +2*60 +2*6=3600+360+120+12=4092. Yes, that's right. So the quadratic factors as (x -66)(x +62)=0. Therefore, x=66 or x=-62. Correct. 

Therefore, the positive solution is 66. 

I think that's thorough. Checked both solutions, verified the calculations, even factored the quadratic. Everything seems to check out. So I'm confident that x=66 is the correct answer.
Final Answer
The positive value of $x$ is $\boxed{66}$.
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