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# Finding all complex fourth roots

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Find all complex fourth-roots in rectangular form of

w= 121 ( cos 2pi/3 + i sin 2pi/3 )

Type answer in the form a + bi round to the nearest tenth.

Zsub 0 = ? + ?i
Zsub 1 = -? + ?i
Zsub 2 = -? - ?i
Zsub 3 = ? - ?i

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https://brainmass.com/math/complex-analysis/demoivres-theorem-complex-roots-443657

#### Solution Preview

If we write the solution of z^4 = w as:

z = r [cos(theta) + i sin(theta)]

then the equation becomes:

r^4 [cos(4 theta) + i sin(4 theta) ] = 121 [ cos(2pi/3) + i sin(2pi/3) ]

This then implies that:

r = 121^(1/4) = sqrt(11)

and

4 theta = 2 ...

#### Solution Summary

We show how DeMoivre's Theorem can be used to compute all the fourth roots of a number.

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