Please help me with this sum below?

enter image source here

1 Answer
Dec 5, 2017

The answer is option #A=16+16sqrt3i#

Explanation:

Let #z=(1/4+sqrt3/4i)#

Convert #z# to the trigonometric form

#z=1/2(1/2+sqrt3/2i)=1/2(costheta+isintheta)#

#costheta=1/2# and #sintheta=sqrt3/2#

#theta# is in the first quadrant and

#theta=1/3pi#

Therefore,

#z=1/2(cos(1/3pi)+isin(1/3pi))=#

Applying Demoivres Theorem

#z^n=(cos(theta)+isin(theta))^n=cos(ntheta)+isin(ntheta)#

Therefore,

#z^(-5)=(1/2(cos(1/3pi)+isin(1/3pi)))^(-5)#

#=2^5(cos(-5/3pi)+isin(-5/3pi))#

#=32(1/2+sqrt3/2i)#

#=16+16sqrt3i#