How do you find the complex cube roots of 8?

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To find the complex cube roots of 8, we can start by expressing the number 8 in polar form. The number 8 can be written as:

8 = 8 + 0i

In polar coordinates, this is represented as:

8 = 8 (cos(0) + i sin(0))

Here, the modulus (r) is 8, and the argument (θ) is 0. To find the cube roots, we will use the formula for finding the n-th roots of a complex number:

w_k = r^{1/n} (cos((θ + 2kπ) / n) + i sin((θ + 2kπ) / n))

Where:

  • n is the order of the root (in this case, n = 3 for cube roots),
  • k is an integer that will take on the values 0, 1, and 2 for the 3 cube roots.

Substituting our values into the formula:

  • r = 8, so r(1/3) = 8(1/3) = 2.
  • θ = 0, so (0 + 2kπ) / 3 = (2kπ) / 3.

Now we can find each of the cube roots by substituting the values of k:

  1. For k = 0:
    • w0 = 2 (cos(0) + i sin(0)) = 2 (1 + 0i) = 2.
  2. For k = 1:
    • w1 = 2 (cos(2π/3) + i sin(2π/3)) = 2 (-1/2 + i√3/2) = -1 + i√3.
  3. For k = 2:
    • w2 = 2 (cos(4π/3) + i sin(4π/3)) = 2 (-1/2 – i√3/2) = -1 – i√3.

Thus, the three complex cube roots of 8 are:

  • w0 = 2
  • w1 = -1 + i√3
  • w2 = -1 – i√3

These roots can be confirmed by raising each of them back to the third power, and they will yield the original number, 8.

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