Solution of x⁵-1=0

Solution to x⁵ - 1 = 0

To solve the equation x⁵ - 1 = 0, we can rewrite it as:

x⁵ = 1

This is a complex equation with roots that can be represented as the fifth roots of unity:

1. Rewrite 1 in complex exponential form:
   1 = e^2πik, where k = 0, 1, 2, 3, 4
2. Taking the fifth root of both sides:
   x = e^{(2πi * k)/5}, where k = 0, 1, 2, 3, 4

The five complex solutions are:

• x₀ = e⁰ = 1
• x₁ = e^{2πi / 5}
• x₂ = e^{4πi / 5}
• x₃ = e^{6πi / 5}
• x₄ = e^{8πi / 5}

Each of these values is a complex root that satisfies x⁵ = 1.