At first, we rewrite z=1+i3 in the polar form. We get: z=2(21+i23)=2(cos(3π)+isin(3π)).
We use the known formula (extension of de Moivre's formula) to compute the root 31. We get:
z31=231(cos(33π+2πk)+isin(33π+2πk)), where k=0,1,2.
We receive the following roots:
v0=231(cos(9π)+isin(9π)),
v1=231(cos(97π)+isin(97π)),
v2=231(−cos(94π)−isin(94π)).
Answer: the cube root of z has the following values:
v0=231(cos(9π)+isin(9π)),
v1=231(cos(97π)+isin(97π)),
v2=231(−cos(94π)−isin(94π)).
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