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z^n = K (K is real).The sum of the arguments of all the roots of this equation is an odd multiple of pi..When is this possible? That is the relation between n and k.

z^n = K (K is real).The sum of the arguments of all the roots of this equation is an odd multiple of pi..When is this possible? That is the relation between n and k.

Grade:Upto college level

1 Answers

AskiitiansExpert Mohit-IITD
21 Points
10 years ago

Dear Aniket,

The sum of all the roots of zn=K is an odd multiple of π when K is negative.

Roots of this equation for K<0 will be z=|K|1/nei(2m+1)π/n where m=0,1,2,...,n-1. Clearly the sum of roots is an odd multiple of π.

e.g. z3=-1

=> z=eiπ/3,e,ei5π/3

Sum of arguments = 3π

Hope this helps.

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