Express the following in the form abicomplex number Express

Express the following in the form a+bi.(complex number)

Express the following in the form a+bi.(complex number) (1 - i)^6 (Squareroot 3 + i)^3

Solution

(1 - i)^6( sqrt3 +i)^3

Convert rectangular to polar form :

(1 -i) = sqrt2{e^i(3pi/4)

(sqrt3 +i) = 2e^ipi/6

Use DeMoivre\'s Theorem:

(1 - i)^6( sqrt3 +i)^3 = [sqrt2{e^i(3pi/4)]^6*[ 2e^ipi/6]^3

= [(2)^3e^i(9pi/2)][2^3e^ipi/2]

=2^6[e^i(9pi/2+pi/2)

=2^6[e^i5pi]

= 2^6[ cos5pi + i*sin5pi ]

= 2^6[ -1]

= - 64