My Coding > Mathematics > Complex numbers > Root of power 8 of 1

Root of power 8 of 1

In order to calculate the root of power 8 from 1 it is necessary to use exponential, or Eulers form if 1. This is the easiest way to find a solution.

From Euler's formula we have \(1 = e^{i(0 + 2\pi*n)}\), so we can write:

\[\sqrt[8]{1} = 1^{\frac{1}{8}} = e^{\frac{1}{8}i(0 + 2\pi*n)} = e^{i(0 + \frac{\pi*n}{4})}\]

Job done! But to understand this answer, we need to write all roots one by one, to expand the sense of the above formula. To do it we need to expand all values within \([0, 2*\pi]\) range, all eight roots.

\[\sqrt[8]{1} = \pm{1}; \pm{i}; \pm{\frac{1}{\sqrt{2}}} + \frac{\pm{i}}{\pm{\sqrt{2}}}\]

Or in a bit simplified form:

\[\sqrt[8]{1} = \pm{1}; \pm{i}; \pm{\frac{1}{\sqrt{2}}}(1 \pm{i})\]

For graphical explanation and some comments please check this video:


Published: 2023-05-08 16:03:27
Updated: 2023-05-08 16:09:45

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