Express the following complex numbers in the $x+iy$form. Try to visualize each complex number, using sketches as in the examples if necessary. The first twelve problems you should be able to do in your head (and maybe some of the others—try it!) Doing a problem quickly in your head saves time over using a computer. Remember that the point in doing problems like this is to gain skill in manipulating complex expressions, so a good study method is to do the problems by hand and use a computer to check your answers.

13.$\frac{{(i-\sqrt{3})}^3}{1-i}$

The given complex number is$\frac{{(i-\sqrt{3})}^3}{1-i}$.

The numbers that are presented in the form of a + ib, where, a,b are real numbers and 'i' is an imaginary number called complex numbers.

Let the given complex number be $\frac{{(i-\sqrt{3})}^3}{1-i}$.

Let z₁ and z₂ be 1 - i and$1-\sqrt{3}$respectively.

The polar form ofz₁ is $z_1=r_1{e}^{\left(i{\theta }_1\right)}$.

The polar form ofz₂ is $z_2=r_2{e}^{\left(i{\theta }_2\right)}$.

Calculate the value of r₁, and${\theta }_1$ as:

$\begin{array}{l}{r}_1=\sqrt{1+3}\\ r_1=2\end{array}$

And,

$\begin{array}{c}{\theta }_1=-\arctan \left(\frac{1}{\sqrt{3}}\right)\\ {\theta }_1=-\frac{\pi }{6}\\ {\theta }_1=\pi -\frac{\pi }{6}\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\left\{\because {\theta }_1\text{lies\hspace{0.17em}in\hspace{0.17em}I\hspace{0.17em}quadrant}\right\}\\ {\theta }_1=\frac{5\pi }{6}\end{array}$

Calculate the value of$r_2$ and${\theta }_2$ as:

$\begin{array}{l}{r}_2=\sqrt{1+1}\\ r_2=\sqrt{2}\\ {\theta }_2=-\arctan \left(\frac{1}{1}\right)\\ {\theta }_2=-\frac{\pi }{4}\end{array}$

Solve further:

$\begin{array}{l}{\theta }_2=2\pi -\frac{\pi }{4}\left\{\because {\theta }_2\text{lies\hspace{0.17em}in\hspace{0.17em}IV\hspace{0.17em}quadrant}\right\}\\ {\theta }_2=\frac{7\pi }{4}\end{array}$

The values of z₁ and z₂ are$2e^{(5\pi i/6)},\sqrt{2}{e}^{(7\pi i/4)}$ respectively.

Calculate the value of zas:

$\begin{array}{l}z=\frac{z_1}{z_2}\\ z=\frac{{\left[2e^{(5\pi i/6)}\right]}^3}{\sqrt{2}{e}^{(7\pi i/4)}}\\ z=4\sqrt{2}{e}^{\left[\right(5\pi i/2)-(7\pi i/4\left)\right]}\\ z=4\sqrt{2}{e}^{[3\pi i/4]}\end{array}$

The value of r and $\theta$in z are $4\sqrt{2},\frac{3\pi }{4}$respectively.

$\begin{array}{l}z=r\left(\cos \left(\theta \right)+i\sin \left(\theta \right)\right)\\ z=4\sqrt{2}\left(\cos \left(\frac{3\pi }{4}\right)+i\sin \left(\frac{3\pi }{4}\right)\right)\\ z=4\sqrt{2}\left(\left(\frac{-1}{\sqrt{2}}\right)+i\left(\frac{1}{\sqrt{2}}\right)\right)\\ z=-4+4i\end{array}$

The general form is$z=-4+4i$

Plot the complex number $z=-4+4i$.

Therefore, the general form is $z=-4+4i$.