Find all the real and complex roots of 8.

Expert Answers

An illustration of the letter 'A' in a speech bubbles

Let the complex number `z=8` . Rewrite this in polar form as `z=8e^(i(0+2pi*n))` .

`z^(1/3)=(8e^(i(0+2pi*n)))^(1/3)`

`z^(1/3)=2e^(i(0+(2pi*n)/3))`

Here, `theta` becomes 3 distinct angles for all integers `n` . `0^@` , `120^@` , and `240^@` . The value at `0^@` is the real root, and the other two are complex roots. These can be represented as equal points around a circle of radius `2` in the complex plane.

The roots in cartesian coordinates are

`8^(1/3)={2, -1+i sqrt(3), -1-i sqrt(3)}`

See eNotes Ad-Free

Start your 48-hour free trial to get access to more than 30,000 additional guides and more than 350,000 Homework Help questions answered by our experts.

Get 48 Hours Free Access
Images:
Image (1 of 1)
Approved by eNotes Editorial