Explanation:
Okay, so I need to solve this equation: the cube root of (x squared minus 4x plus 4) equals 16. Let me write that down first to make sure I have it right.

∛(x² - 4x + 4) = 16

Alright, so the cube root of something is 16. To get rid of the cube root, I think I need to cube both sides of the equation. Let me try that.

If I cube both sides, then the left side becomes just the inside of the cube root, right? Because (³√a)³ is a. So that would leave me with:

x² - 4x + 4 = 16³

Hmm, okay. Let me compute 16 cubed. 16 times 16 is 256, and then 256 times 16... Let me do that step by step. 256 * 10 is 2560, 256 * 6 is 1536, so adding those together: 2560 + 1536 = 4096. So 16³ is 4096. Therefore:

x² - 4x + 4 = 4096

Now, I need to solve this quadratic equation. Let me rearrange it to standard form. Subtract 4096 from both sides:

x² - 4x + 4 - 4096 = 0

Which simplifies to:

x² - 4x - 4092 = 0

Wait, is that right? Let me check. 4 - 4096 is indeed -4092. Yes. So the quadratic is x² - 4x - 4092 = 0.

Now, how do I solve this quadratic equation? Well, I can try factoring, but given the numbers, it might not be straightforward. Let me check the discriminant first to see if it factors nicely or if I need to use the quadratic formula.

The discriminant D is b² - 4ac. Here, a = 1, b = -4, c = -4092.

Calculating D: (-4)² - 4*(1)*(-4092) = 16 + 4*4092.

Let me compute 4*4092. Let's break it down: 4*4000 = 16000, and 4*92 = 368, so total is 16000 + 368 = 16368. Then adding 16 gives D = 16368 + 16 = 16384.

Wait, 16384? Hmm, that number seems familiar. Isn't that 128 squared? Let me check. 128*128. 100*128=12800, 20*128=2560, 8*128=1024. Adding them: 12800 + 2560 = 15360, plus 1024 is 16384. Yes! So sqrt(16384) is 128. That's good because that means the roots will be rational.

So using quadratic formula:

x = [4 ± √D]/2. Wait, but the formula is x = [-b ± √D]/(2a). Since a = 1, b = -4, it's:

x = [4 ± √16384]/2

Which is:

x = [4 ± 128]/2

So that gives two solutions:

First, 4 + 128 = 132. Divided by 2 is 66.

Second, 4 - 128 = -124. Divided by 2 is -62.

So the solutions are x = 66 and x = -62. But the problem asks for the positive value of x, so the answer is 66. Let me check if this is correct.

Let me verify by plugging x = 66 back into the original equation.

Compute the expression inside the cube root first: x² -4x +4.

x² = 66². Let me calculate that. 60² = 3600, 6² = 36, and then the cross term 2*60*6 = 720. So (60 + 6)² = 60² + 2*60*6 + 6² = 3600 + 720 + 36 = 4356.

Then, -4x is -4*66 = -264. Then +4. So altogether:

4356 - 264 + 4. Let me compute 4356 - 264 first. 4356 - 200 = 4156, then -64 more is 4156 - 64 = 4092. Then adding 4 gives 4096. So the expression inside the cube root is 4096, and ∛4096 is indeed 16, because 16³ = 4096. So that checks out.

Now, let me check the negative solution x = -62 just to be thorough.

x² = (-62)² = 3844.

-4x is -4*(-62) = +248. Then +4. So total inside the cube root is 3844 + 248 + 4.

Compute 3844 + 248: 3844 + 200 = 4044, +48 = 4092. Then +4 is 4096. Same as before. So ∛4096 = 16. So both solutions satisfy the equation. However, the problem asks for the positive value of x, so 66 is the answer.

Wait, but hold on a second. The original equation has a cube root. Cube roots can handle negative numbers, right? So even if the inside was negative, the cube root would still be defined. However, in this case, both solutions make the inside of the cube root positive, so both are valid. But since the problem asks for the positive value, 66 is the answer.

But just to make sure I didn't make any mistakes in my calculations. Let me go through each step again.

Original equation: ∛(x² -4x +4) = 16.

Cubing both sides: x² -4x +4 = 16³ = 4096. Then moving 4096 over: x² -4x -4092 = 0. Discriminant D = 16 + 4*4092 = 16 + 16368 = 16384. sqrt(16384) = 128. Then solutions are [4 ± 128]/2. So 132/2 = 66, and -124/2 = -62. Then checked both solutions. Both work. So positive solution is 66. All steps seem okay.

Alternatively, maybe I can factor the quadratic equation. Let me try. The quadratic is x² -4x -4092 = 0. If I try to factor it, I need two numbers that multiply to -4092 and add to -4. Hmm. Let's see.

Alternatively, maybe I can write the original expression inside the cube root. Wait, x² -4x +4. Wait a second, isn't that a perfect square? Let me check.

x² -4x +4. Let me see: (x - 2)² = x² -4x +4. Yes! Oh! So the original equation is ∛[(x - 2)²] = 16. So maybe that can be another way to approach it?

So instead of expanding, maybe I can write:

∛[(x - 2)²] = 16. Then cube both sides: (x - 2)² = 16³ = 4096. Then take square roots: x - 2 = ±√4096. Since √4096 is 64, because 64² = 4096. Wait, 64*64 is 4096? Let me check. 60² = 3600, 4² = 16, 2*60*4 = 480, so (60 + 4)² = 3600 + 480 + 16 = 4096. Yes! So √4096 is 64. Therefore, x - 2 = ±64.

Therefore, solving for x:

x - 2 = 64 => x = 66

or

x - 2 = -64 => x = -62

So same answers. So that's a better way. I didn't notice that x² -4x +4 was a perfect square. So maybe that's a simpler approach. Either way, we get the same solutions.

So, since the problem asks for the positive value of x, it's 66. So I think that's the answer. Let me just check once more with the original equation.

Original equation: ∛[(x - 2)^2] = 16. For x = 66, (66 - 2)^2 = 64² = 4096. ∛4096 = 16. Correct. For x = -62, (-62 - 2)^2 = (-64)^2 = 4096, same thing. So both work. Positive x is 66. So I think that's solid.

Therefore, the positive value of x is 66.

**Final Answer**
The positive value of $ x $ is \boxed{66}.

To solve the equation:

$$
\sqrt[3]{x^2 - 4x + 4} = 16
$$

we begin by isolating the radical. Since the left-hand side is a cube root, we can eliminate it by cubing both sides of the equation:

$$
\left(\sqrt[3]{x^2 - 4x + 4}\right)^3 = 16^3
$$

This simplifies to:

$$
x^2 - 4x + 4 = 4096
$$

Now observe that the expression inside the cube root, $ x^2 - 4x + 4 $, is a perfect square:

$$
x^2 - 4x + 4 = (x - 2)^2
$$

So we can rewrite the equation as:

$$
\sqrt[3]{(x - 2)^2} = 16
$$

Cube both sides again:

$$
(x - 2)^2 = 16^3 = 4096
$$

Now take the square root of both sides:

$$
x - 2 = \pm \sqrt{4096}
$$

We compute the square root:

$$
\sqrt{4096} = 64
$$

So we have two cases:

$$
x - 2 = 64 \quad \text{or} \quad x - 2 = -64
$$

Solving for $ x $ in each case:

- $ x = 64 + 2 = 66 $
- $ x = -64 + 2 = -62 $

We are asked to find the **positive** value of $ x $, so the solution is:

$$
\boxed{66}
$$

[Assistant]
Explanation:
Okay, so I need to solve this equation: the cube root of (x squared minus 4x plus 4) equals 16. Let me write that down first to make sure I have it right.

∛(x² - 4x + 4) = 16

Alright, so the cube root of something is 16. To get rid of the cube root, I think I need to cube both sides of the equation. Let me try that.

If I cube both sides, then the left side becomes just the inside of the cube root, right? Because (³√a)³ is a. So that would leave me with:

x² - 4x + 4 = 16³

Hmm, okay. Let me compute 16 cubed. 16 times 16 is 256, and then 256 times 16... Let me do that step by step. 256 * 10 is 2560, 256 * 6 is 1536, so adding those together: 2560 + 1536 = 4096. So 16³ is 4096. Therefore:

x² - 4x + 4 = 4096

Now, I need to solve this quadratic equation. Let me rearrange it to standard form. Subtract 4096 from both sides:

x² - 4x + 4 - 4096 = 0

Which simplifies to:

x² - 4x - 4092 = 0

Wait, is that right? Let me check. 4 - 4096 is indeed -4092. Yes. So the quadratic is x² - 4x - 4092 = 0.

Now, how do I solve this quadratic equation? Well, I can try factoring, but given the numbers, it might not be straightforward. Let me check the discriminant first to see if it factors nicely or if I need to use the quadratic formula.

The discriminant D is b² - 4ac. Here, a = 1, b = -4, c = -4092.

Calculating D: (-4)² - 4*(1)*(-4092) = 16 + 4*4092.

Let me compute 4*4092. Let's break it down: 4*4000 = 16000, and 4*92 = 368, so total is 16000 + 368 = 16368. Then adding 16 gives D = 16368 + 16 = 16384.

Wait, 16384? Hmm, that number seems familiar. Isn't that 128 squared? Let me check. 128*128. 100*128=12800, 20*128=2560, 8*128=1024. Adding them: 12800 + 2560 = 15360, plus 1024 is 16384. Yes! So sqrt(16384) is 128. That's good because that means the roots will be rational.

So using quadratic formula:

x = [4 ± √D]/2. Wait, but the formula is x = [-b ± √D]/(2a). Since a = 1, b = -4, it's:

x = [4 ± √16384]/2

Which is:

x = [4 ± 128]/2

So that gives two solutions:

First, 4 + 128 = 132. Divided by 2 is 66.

Second, 4 - 128 = -124. Divided by 2 is -62.

So the solutions are x = 66 and x = -62. But the problem asks for the positive value of x, so the answer is 66. Let me check if this is correct.

Let me verify by plugging x = 66 back into the original equation.

Compute the expression inside the cube root first: x² -4x +4.

x² = 66². Let me calculate that. 60² = 3600, 6² = 36, and then the cross term 2*60*6 = 720. So (60 + 6)² = 60² + 2*60*6 + 6² = 3600 + 720 + 36 = 4356.

Then, -4x is -4*66 = -264. Then +4. So altogether:

4356 - 264 + 4. Let me compute 4356 - 264 first. 4356 - 200 = 4156, then -64 more is 4156 - 64 = 4092. Then adding 4 gives 4096. So the expression inside the cube root is 4096, and ∛4096 is indeed 16, because 16³ = 4096. So that checks out.

Now, let me check the negative solution x = -62 just to be thorough.

x² = (-62)² = 3844.

-4x is -4*(-62) = +248. Then +4. So total inside the cube root is 3844 + 248 + 4.

Compute 3844 + 248: 3844 + 200 = 4044, +48 = 4092. Then +4 is 4096. Same as before. So ∛4096 = 16. So both solutions satisfy the equation. However, the problem asks for the positive value of x, so 66 is the answer.

Wait, but hold on a second. The original equation has a cube root. Cube roots can handle negative numbers, right? So even if the inside was negative, the cube root would still be defined. However, in this case, both solutions make the inside of the cube root positive, so both are valid. But since the problem asks for the positive value, 66 is the answer.

But just to make sure I didn't make any mistakes in my calculations. Let me go through each step again.

Original equation: ∛(x² -4x +4) = 16.

Cubing both sides: x² -4x +4 = 16³ = 4096. Then moving 4096 over: x² -4x -4092 = 0. Discriminant D = 16 + 4*4092 = 16 + 16368 = 16384. sqrt(16384) = 128. Then solutions are [4 ± 128]/2. So 132/2 = 66, and -124/2 = -62. Then checked both solutions. Both work. So positive solution is 66. All steps seem okay.

Alternatively, maybe I can factor the quadratic equation. Let me try. The quadratic is x² -4x -4092 = 0. If I try to factor it, I need two numbers that multiply to -4092 and add to -4. Hmm. Let's see.

Alternatively, maybe I can write the original expression inside the cube root. Wait, x² -4x +4. Wait a second, isn't that a perfect square? Let me check.

x² -4x +4. Let me see: (x - 2)² = x² -4x +4. Yes! Oh! So the original equation is ∛[(x - 2)²] = 16. So maybe that can be another way to approach it?

So instead of expanding, maybe I can write:

∛[(x - 2)²] = 16. Then cube both sides: (x - 2)² = 16³ = 4096. Then take square roots: x - 2 = ±√4096. Since √4096 is 64, because 64² = 4096. Wait, 64*64 is 4096? Let me check. 60² = 3600, 4² = 16, 2*60*4 = 480, so (60 + 4)² = 3600 + 480 + 16 = 4096. Yes! So √4096 is 64. Therefore, x - 2 = ±64.

Therefore, solving for x:

x - 2 = 64 => x = 66

or

x - 2 = -64 => x = -62

So same answers. So that's a better way. I didn't notice that x² -4x +4 was a perfect square. So maybe that's a simpler approach. Either way, we get the same solutions.

So, since the problem asks for the positive value of x, it's 66. So I think that's the answer. Let me just check once more with the original equation.

Original equation: ∛[(x - 2)^2] = 16. For x = 66, (66 - 2)^2 = 64² = 4096. ∛4096 = 16. Correct. For x = -62, (-62 - 2)^2 = (-64)^2 = 4096, same thing. So both work. Positive x is 66. So I think that's solid.

Therefore, the positive value of x is 66.

**Final Answer**
The positive value of $ x $ is \boxed{66}.

To solve the equation:

$$
\sqrt[3]{x^2 - 4x + 4} = 16
$$

we begin by isolating the radical. Since the left-hand side is a cube root, we can eliminate it by cubing both sides of the equation:

$$
\left(\sqrt[3]{x^2 - 4x + 4}\right)^3 = 16^3
$$

This simplifies to:

$$
x^2 - 4x + 4 = 4096
$$

Now observe that the expression inside the cube root, $ x^2 - 4x + 4 $, is a perfect square:

$$
x^2 - 4x + 4 = (x - 2)^2
$$

So we can rewrite the equation as:

$$
\sqrt[3]{(x - 2)^2} = 16
$$

Cube both sides again:

$$
(x - 2)^2 = 16^3 = 4096
$$

Now take the square root of both sides:

$$
x - 2 = \pm \sqrt{4096}
$$

We compute the square root:

$$
\sqrt{4096} = 64
$$

So we have two cases:

$$
x - 2 = 64 \quad \text{or} \quad x - 2 = -64
$$

Solving for $ x $ in each case:

- $ x = 64 + 2 = 66 $
- $ x = -64 + 2 = -62 $

We are asked to find the **positive** value of $ x $, so the solution is:

$$
\boxed{66}
$$

Solution:
We first cube each side of the equation to get $x^2 - 4x + 4 = 16^3$.  Notice that $x^2 - 4x + 4 = (x-2)^2.$

Therefore, we have that $x-2 = \pm 16^{3/2} = \pm 64$.  Therefore, the possible values of $x$ are $-62$ and $66,$ and the only positive value is therefore $\boxed{66}$.