Question

Determine the sixth roots of 8 as well as of -8

Answer 1

for 8, sqrt(2),
for -8, -sqrt(2)

Answer 2

you want the six complex roots right?

Answer 3

yes the six complex roots

Answer 4

lets do it the snap way
draw the unit circle. then we can easily find the sixth roots of 1
once we have those, we multiply each by
\[\sqrt{2}\] because
\[\sqrt{2}^6=2^3=8\]

Answer 5

since we know that
\[1^6=1\] we know one sixth root of 1, namely 1. divide the unit circle into six equal parts, where one is at (1,0)

Answer 6

working in degrees you can say 360/6=60 or in radians
\[\frac{2\pi}{6}=\frac{\pi}{3}\] so the coordinates of each of these points should be easy to find. first one is
\[\cos(60)+i\sin(60)=\frac{1}{2}+\frac{\sqrt{3}}{2}i\]

Answer 7

others also easy to find
\[-\frac{1}{2}+\frac{\sqrt{3}}{2}i,-1,-\frac{1}{2}+-\frac{\sqrt{3}}{2}i,\frac{1}{2}-\frac{\sqrt{3}}{2}i\]

Answer 8

multiply each of these by
\[\sqrt{2}\] and you are done

Answer 9

Oh, i thought he just wanted 8^(1/6) hahaha