Algebra

To solve the equation z + 1/z = √2, we first express z in polar form as z = re^(iθ), where r is the modulus and θ is the argument. This is a common method in complex number analysis.

When we substitute z into the equation, we get: re^(iθ) + 1/(r*e^(iθ)) = √2 This transforms into: r²e^(iθ) + 1 = r√2

By separating the real and imaginary parts, we can find valuable insights about r and θ.

Finding r and θ From the equation r² sin(θ) = 0, we conclude that sin(θ) = 0. This suggests that θ can either be 0 or π (or any multiple of π).

If n is even and cos(θ) = 1: r² + 1 = r√2 leads to the quadratic equation: r² - r√2 + 1 = 0

Solving this using the quadratic formula gives r = 1.

If n is odd and cos(θ) = -1: -r² + 1 = r√2 results in a different quadratic equation, but it does not yield a positive r.

Determining the Value of z Since the positive modulus is verified to be r = 1, we find z = e^(iθ). Thus:

For θ = 0, z = 1.

For θ = π, z = -1.

Calculating z^10 + 1/z^10 For z = 1: 1^10 + 1/1^10 = 1 + 1 = 2.

For z = -1: (-1)^10 + 1/(-1)^10 = 1 + 1 = 2.

So the Final answer is 2.