Given-that-one-of-the-value-for-the-5-th-root-of-a-complex-number-is-1-i-Find-the-another-four-values-

Question Number 136167 by ZiYangLee last updated on 19/Mar/21

Given that one of the value for the 5^{th}

root of a complex number is - 1 + i .

Find the another four values .

Answered by mr W last updated on 19/Mar/21

o n e 5^{t h} r o o t i s - 1 + i = \sqrt{2} (\cos \frac{3 π}{4} + i \sin \frac{3 π}{4})

\frac{3 π}{4} + \frac{2 π}{5} = \frac{23 π}{20}

\frac{3 π}{4} + \frac{2 π}{5} + \frac{2 π}{5} = \frac{31 π}{20}

\frac{3 π}{4} + \frac{2 π}{5} + \frac{2 π}{5} + \frac{2 π}{5} = \frac{39 π}{20}

\frac{3 π}{4} - \frac{2 π}{5} = \frac{7 π}{20}

t h e o t h e r 5^{t h} r o o t s a r e :

\sqrt{2} [\cos (\frac{7 π}{20}) + i \sin (\frac{7 π}{20})]

\sqrt{2} [\cos (\frac{23 π}{20}) + i \sin (\frac{23 π}{20})]

\sqrt{2} [\cos (\frac{31 π}{20}) + i \sin (\frac{31 π}{20})]

\sqrt{2} [\cos (\frac{39 π}{20}) + i \sin (\frac{39 π}{20})]

Commented by mr W last updated on 19/Mar/21

Commented by mr W last updated on 20/Mar/21

a k n o w n n^{t h} r o o t o f z

t h e o t h e r n^{t h} r o o t s o f z

i f o n e o f t h e n^{t h} r o o t s o f a n u m b e r i s

{r e}^{α i}, t h e n t h e o t h e r n^{t h} r o o t s o f i t a r e

{r e}^{(α + \frac{2 k π}{n}) i} w i t h k = 1, 2, \dots, n - 1

Commented by ZiYangLee last updated on 20/Mar/21

Thanks Sir!