Find the three cube roots of 1

To find the three cube roots of 1, we start with the equation:

x³ = 1

This means we are looking for the values of x that satisfy this equation. One obvious solution is x = 1. However, because this is a cubic equation, there can be other solutions as well. To find the other cube roots, we can use the concept of complex numbers and polar coordinates.

1 can be expressed in polar form as:

1 = cos(0) + i sin(0),

which is equivalent to a modulus of 1 and an argument (angle) of 0 radians.

Using De Moivre’s Theorem, the cube roots can be found by dividing the angle by 3 and adding multiples of (2π/3).

The formula for the k-th cube root of a complex number is:

x_k = r^1/3(cos((θ + 2kπ) / 3) + i sin((θ + 2kπ) / 3)),

where k = 0, 1, 2, …

Here, r = 1 (the modulus) and θ = 0 (the argument). Let’s calculate the three roots:

• For k = 0:
• x₀ = cos(0) + i sin(0) = 1
• For k = 1:
• x₁ = cos(2π/3) + i sin(2π/3) = -1/2 + i(√3/2)
• For k = 2:
• x₂ = cos(4π/3) + i sin(4π/3) = -1/2 – i(√3/2)

Therefore, the three cube roots of 1 are:

1. x = 1
2. x = -1/2 + i(√3/2)
3. x = -1/2 – i(√3/2)

These roots represent the three distinct cube roots of the number 1 in the complex plane.