What is complex conjugate of [ (3 + 8i)^4 ] / [ (1+i)^10 ](3+8i)4(1+i)10?

1 Answer
Jan 10, 2016

-165+721/32i165+72132i

Explanation:

First note that (1+i)^2 = 1^2+2i+i^2 = 2i(1+i)2=12+2i+i2=2i and (2i)^2 = -4(2i)2=4

So

(1+i)^10 = (2i)^5 = (2i)^2(2i)^2(2i) = (-4)(-4)(2i) = 32i(1+i)10=(2i)5=(2i)2(2i)2(2i)=(4)(4)(2i)=32i

(3+8i)^4 = (9+48i+64i^2)^2 = (-55+48i)^2(3+8i)4=(9+48i+64i2)2=(55+48i)2

= (3025-5280i+2304i^2) = 721-5280i=(30255280i+2304i2)=7215280i

So:

(3+8i)^4/(1+i)^10 = (721-5280i)/(32i) = -721/32i-5280/32 = -721/32i-165 = -165-721/32i(3+8i)4(1+i)10=7215280i32i=72132i528032=72132i165=16572132i

So, reversing the sign of the coefficient of ii, the Complex conjugate is:

-165+721/32i165+72132i