Question 1116764
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Put the numbers in {{{e^(ix)}}} format and use deMoivre's Theorem.<br>
deMoivre's Theorem says that to find the n-th root of a complex number, you take the n-th root of the modulus and divide the angle by n.<br>
{{{cos(x)+i*sin(x) = cis(x) = e^(ix)}}}<br>
For your problem, we are to find the values of {{{8^(1/3)}}}.  We have<br>
{{{8 = 8e^(i*0) = 8e^(i*2pi) = 8e^(i*4pi)}}<br>
Then<br>
{{{8^(1/3) = (8)^(1/3)*e^(i*0/3) = (8)^(1/3)*e^(i*2pi/3) = (8)^(1/3)*e^(i*4pi/3)}}}<br>
The three cube roots of 8 are
(1) {{{2e^i(0) = 2cis(0) = 2}}}
(2) {{{2e^(i(2pi/3)) = 2cis(2pi/3) = -1/2+i*sqrt(3)/2}}}
(3) {{{2e^(i(4pi/3)) = 2cis(4pi/3) = -1/2-i*sqrt(3)/2}}}