Okay, let's see. The problem is asking me to compute the value of z + 1/z + z² + 1/z², where z is a complex number such that z⁵ = 1 and z ≠ 1. Hmm. So z is a 5th root of unity, excluding 1. The 5th roots of unity are the solutions to the equation z⁵ = 1, right? They are usually expressed in the complex plane as points on the unit circle spaced at angles of 72 degrees apart, starting from 1 (which is at 0 degrees). So excluding 1, there are four other roots. But maybe I don't need to think about them geometrically right away.

First, let me recall that for any root of unity, z ≠ 1, we have the property that 1 + z + z² + z³ + z⁴ = 0. That's because z⁵ - 1 = 0 factors as (z - 1)(1 + z + z² + z³ + z⁴) = 0, and since z ≠ 1, the sum 1 + z + z² + z³ + z⁴ must be zero. Maybe that can help here.

But the problem isn't directly asking for the sum of z, z², etc., but rather the sum of z + 1/z + z² + 1/z². Let me see. Since z is a root of unity, 1/z is equal to z⁴ because z⁵ = 1 implies z⁴ = 1/z. Similarly, 1/z² = z³. So maybe I can rewrite the expression in terms of lower powers of z.

Let me try that. Let's compute each term:

1/z = z⁴ (since z⁵ = 1, so dividing both sides by z gives z⁴ = 1/z)
Similarly, 1/z² = z³ (since z⁵ = 1, so z³ = 1/z²)

Therefore, the expression z + 1/z + z² + 1/z² becomes z + z⁴ + z² + z³. So that's z + z² + z³ + z⁴. But from the earlier equation, we know that 1 + z + z² + z³ + z⁴ = 0. Therefore, z + z² + z³ + z⁴ = -1. So substituting back, the expression we need to compute is -1. Wait, is that possible?

Wait, let me verify that again. The original equation is 1 + z + z² + z³ + z⁴ = 0. Therefore, subtracting 1 from both sides gives z + z² + z³ + z⁴ = -1. So indeed, the sum z + z² + z³ + z⁴ is -1. But in our problem, we have z + 1/z + z² + 1/z² = z + z⁴ + z² + z³ = z + z² + z³ + z⁴ = -1. So is the answer simply -1? That seems straightforward.

Wait, but let me check with a specific root of unity to make sure. Let's take z = e^(2πi/5), one of the primitive 5th roots of unity. Then 1/z = e^(-2πi/5) = e^(8πi/5), which is z⁴. Similarly, 1/z² = e^(-4πi/5) = e^(6πi/5) = z³. Therefore, z + 1/z = z + z⁴, and z² + 1/z² = z² + z³. So adding those together gives z + z² + z³ + z⁴ = -1. So that checks out.

Alternatively, let's compute numerically. Let me take z = e^(2πi/5). Then compute z + 1/z + z² + 1/z² numerically.

First, z = cos(72°) + i sin(72°) ≈ 0.3090 + 0.9511i
1/z = e^(-2πi/5) ≈ cos(72°) - i sin(72°) ≈ 0.3090 - 0.9511i
z² = e^(4πi/5) ≈ cos(144°) + i sin(144°) ≈ -0.8090 + 0.5878i
1/z² = e^(-4πi/5) ≈ cos(144°) - i sin(144°) ≈ -0.8090 - 0.5878i

Adding all four terms:

z + 1/z ≈ (0.3090 + 0.9511i) + (0.3090 - 0.9511i) = 0.6180
z² + 1/z² ≈ (-0.8090 + 0.5878i) + (-0.8090 - 0.5878i) = -1.6180

Adding those two results: 0.6180 + (-1.6180) = -1.0

So indeed, numerically it also gives -1. Therefore, the answer is -1. So maybe the problem is straightforward once you realize that 1/z = z⁴ and 1/z² = z³, and then use the sum of all roots except 1.

But let me think if there's another way to approach this problem, perhaps using algebraic manipulation or logarithms or something else. For example, maybe writing the expression in terms of cosine, since z = e^(iθ), so 1/z = e^(-iθ), and similarly for z² and 1/z². Then the expression becomes 2cosθ + 2cos2θ. Then maybe compute 2cosθ + 2cos2θ for θ = 72°, 144°, etc. Let's see.

If z is a primitive 5th root of unity, then θ = 72°, 144°, 216°, 288°, and 360°, but 360° is 0°, which is z=1, which is excluded. So the angles are 72°, 144°, 216°, 288°. Let's take θ = 72°, then 2cos72° + 2cos144°. Let's compute that.

cos72° ≈ 0.3090, so 2cos72° ≈ 0.6180
cos144° ≈ -0.8090, so 2cos144° ≈ -1.6180
Adding them: 0.6180 - 1.6180 = -1.0

Similarly, for θ = 144°, we have 2cos144° + 2cos288°. cos288° = cos(360° - 72°) = cos72° ≈ 0.3090. So 2cos144° ≈ -1.6180, 2cos288° ≈ 0.6180. Adding them: -1.6180 + 0.6180 = -1.0. Same result.

Same with θ = 216°, which is cos216° + cos432°. But cos432° = cos(72°) since 432° - 360° = 72°, so cos216° = cos(180° + 36°) = -cos36° ≈ -0.8090, cos432° = cos72° ≈ 0.3090. So 2cos216° + 2cos432° = 2*(-0.8090) + 2*(0.3090) = -1.6180 + 0.6180 = -1.0.

Same with θ = 288°, which is cos288° + cos576°. cos576° = cos(576° - 360°) = cos216° ≈ -0.8090. So 2cos288° + 2cos576° = 2*(0.3090) + 2*(-0.8090) = 0.6180 - 1.6180 = -1.0.

So regardless of which primitive 5th root we take, the expression z + 1/z + z² + 1/z² equals -1. Therefore, the answer is -1. So the problem is solved in two ways: either by recognizing the sum of the roots except 1, or by converting the complex numbers to trigonometric form and computing the sum. Both approaches lead to the same result.

Alternatively, maybe there's a way to use minimal polynomials or field extensions, but that might be overcomplicating it. Since we've confirmed it through multiple methods, I think the answer is solid.

So, putting it all together, the value of the given expression is -1.

**Final Answer**
\boxed{-1}
Therefore, the value of $z + \frac{1}{z} + z^2 + \frac{1}{z^2}$ is $\boxed{-1}$.
</think>