Suppose-z-50-z-25-m-0-where-z-1-i-2-find-the-value-of-m-

Question Number 143194 by ZiYangLee last updated on 11/Jun/21

Suppose z^{50} + z^{25} + m = 0, where z = \frac{1 + i}{\sqrt{2}}

find the value of m .

Answered by Olaf_Thorendsen last updated on 11/Jun/21

z = \frac{1 + i}{\sqrt{2}} = e^{i \frac{π}{4}}

z^{50} = e^{i \frac{π}{4} \times 50} = e^{i \frac{25 π}{2}} = e^{i \frac{π}{2}} = i

z^{25} = e^{i \frac{π}{4} \times 25} = e^{i \frac{25 π}{4}} = e^{i \frac{π}{4}} = \frac{1 + i}{\sqrt{2}} = z

m = - z^{25} - z^{50} = - (z + i) = - \frac{1 + (1 + \sqrt{2}) i}{\sqrt{2}}

Commented by ZiYangLee last updated on 11/Jun/21

thanks sir