What is (2−2i√3)^4 equivalent to? Express the answer in polar form and then in rectangular form.

To find ( 2 - 2i·sqrt(3) )⁴

 

First place the number  2 - 2i·sqrt(3)  into "cis form"

  --     r  =  sqrt( (2)² + [-2·sqrt(3)]² )  =  sqrt( 4 + 12 )  =  sqrt( 16 )  =  4

  --     theta  =  tan^-1( -2sqrt(3) / 2 )  =  5pi/3

So the number can be written as:  4·cis( 5pi/3 ).

 

To find the n^th power of  r·cis( theta)  =  rⁿ·cis( n·theta)

 

So:  ( 2 - 2i·sqrt(3) )⁴  =  4⁴·cis( 4·5pi/3 )  =  256·cis( 20pi/3 ).

 

You'll need to reduce  20pi/3.

 

Then, to put it back into rectangular form, multiply out the cis form.