For any integer n0 find solutions to Can you solve this by u

For any integer n>0, find solutions to:

Can you solve this by using polar coordinates? thank you!

For any integer n>0, find solutions to: z^n = 1 Solve this by using polar coordinates?

Consider the equation : zⁿ = 1

To solve this equation,we use polar representation for both sides,

setting z = reⁱ on the left, and using all possible polar angles on the right;

using the exponential law to multiply , the above equation then becomes rⁿeⁱⁿ = 1.e^{2ki ,} k = 0,±1,±2,±3,...

Equating the absolute values and the polar angles of the two sides gives rⁿ = 1 and n = 2k , k = 0,±1,±2,±3,...

from which we conclude that rⁿ = 1 and = 2k/n ; for n = 0,1,2,3,...,(n-1)

In the above,we get only the value r = 1, since r must be real and non negative.

Thus , we conclude that , the nth root of 1 are the numbers e^2ki/n for k = 0,1,2,...,(n-1)