let z=root3-i and w=-2-2i.
part i: convert z and w to polar form.
part ii: calculate wz using de moivre's theorem. express your answer in polar(r(cos theta+i sin theta)) form.
part iii: calculate z^2. express your answer in rectangular (a+bi) form.
part iv: calculate w^4. express you answer in polar (r(cos theta+i sin theta)) form.

Part I: The complex number √3-i in the polar form will be

The complex number -2-2i in the polar form will be  

Part II: The expression of the product of wz in polar form is

Part III: The expression of z² value in polar form is

Part IV:  The expression of w^4 in polar form is expressed as

Complex numbers are the square roots of any negative numbers. They have both real and imaginary axis.

The rectangular form of expressing complex numbers is expressed as:

The polar representation is expressed as:

Given the complex numbers

z=√3-i

w=-2-2i

Part I: To express in polar form, we need to first get the modulus and argument of each of the complex numbers.

z= √3-i

|z| =

For the argument:

Since tan is negative in the 3rd quadrant

The complex number √3-i in the polar form will be

For the complex number:

z= -2-2i

|z| =

For the argument:

The complex number -2-2i in the polar form will be

Part II: Taking the product of wz

(√3 - 1)(-2-2i)

Expand

wz = -2√3-2√3 i + 2 + 2i

wz = (-2√3+2) +(-2√3+2)i

Get the modulus

Get the argument:

Expression in polar form is

Part III: To calculate z², we will simply square the polar form of z and apply De Moivre's theorem as shown:

Since z =

This shows that the expression of z² value in polar form is

Part 4: We will also use De Moivre's theorem to get w⁴

Since w=

Hence the expression of w^4 in polar form is expressed as

Learn more about complex numbers here: link

answer

step-by-step explanation: