What value in the nth root of unity for any value n?.

\(Z^n=1\\\\Z=(1)^{\frac{1}{n}}\\\\Z=[cos (2k\pi+ 0)+i sin(2k\pi+ 0)]^{\frac{1}{n}}\\\\Z=cos(\frac{2k\pi}{n})+i sin(\frac{2k\pi}{n})=e^{\frac{2k\pi}{n}}\)

Where,k=0,1,2,3......(n-1).

there will be n-roots of the above equation.

For,different values of k,there will be different Z.

\(Z=1,e^{\frac{2\pi}{n}} ,e^{\frac{4\pi}{n}},e^{\frac{6\pi}{n}},e^{\frac{8\pi}{n}},....e^{\frac{2(n-1)\pi}{n}\)

Used,Euler's Identity

\(e^{i\alpha}=cos\alpha + isin\alpha\)

ayoushivt ayoushivt · Answer 2 · 2022-07-18T03:20:10-04:00

For different values of k , the different roots of unity will be

Z = \(\rm 1 , e^{2\pi / n} , \rm e^{4\pi / n}\)....................

What is Euler's Identity ?

According to Euler Identity , \(\rm e^{i\pi}\) = -1

It is asked to determine nth root of Unity or any value of n

Zⁿ = 1

Z = \(\rm 1^{1/n}\)

By Euler's Formula

Z = { cos ( 2kπ +0 ) + isin ( 2kπ +0 )}

Z = cos ( 2kπ / n) + isin ( 2kπ / n)

From euler' identity this can be written as

Z = \(\rm e^{2k\pi / n}\)

where k = 0 , 1 , 2 , 3..... (n-1)

So , the roots of unity will be n different roots.

Therefore , For different values of k , the different roots are as follows

Z = \(\rm 1 , e^{2\pi / n} , \rm e^{4\pi / n}\)....................

To know more about Euler's Identity

https://brainly.com/question/14351184

#SPJ2