If z=\frac{(\sqrt 3+i)^n}{(\sqrt 3-i)^m} , find the relation between m

compagnia04

compagnia04

Answered question

2022-01-16

If z=(3+i)n(3i)m , find the relation between m and n such that z is a real number.

Answer & Explanation

reinosodairyshm

reinosodairyshm

Beginner2022-01-17Added 36 answers

You may continue as follows,
z=2nmcosnπ6+isinnπ6cos11mπ6+isin11mπ6
=2nm(cosnπ6+isinnπ6)(cos11mπ6isin11mπ6)cos211mπ6+sin211mπ6
Then, set the the imaginary part of the numerator to zero,
I=sinnπ6cos11mπ6cosnπ6sin11mπ6=sin(11mn)π6=sin(m+n)π6=0
which leads to (m+n)π6=kπ . Thus, the relationship between m and n is
m+n=6k
with k=0,1,2,

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