Question: Express the following complex numbers in the 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.

25.${\left[\frac{1\sqrt{2}}{1+i}\right]}^{12}$

The given complex number is${\left[\frac{1\sqrt{2}}{1+i}\right]}^{12}$.

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

Let the given number be $z={\left[\frac{1\sqrt{2}}{1+i}\right]}^{12}$ … (1)

The standard form of the complex number is x+iy .

Let$z_1=i\sqrt{2}$.

The modulus of z₁ is$r_1=\sqrt{2}$.

The argument of z₁is localid="1650530563830" ${\theta }_1=arc\tan \left[\frac{\sqrt{2}}{0}\right]$.

${\theta }_1=\frac{\pi }{2}$

The exponential form of the complex number is $r{e}^{i\theta }$.

Hence, $z_1=\sqrt{2}{e}^{(\pi i/2)}$ … (2)

Let $z_2=i+1$.

The modulus of z₂ is $r_2=\sqrt{i+1}$ .

So,$r_2=\sqrt{2}$.

The argument of z₂ is ${\theta }_2=arc\tan \left[\frac{1}{1}\right]$.

So,${\theta }_2=\frac{\pi }{4}$.

Hence $z_2=\sqrt{2}{e}^{(\pi i/4)}$ … (3)

Put the equations (2) and (3) in the equation (1).

$$\begin{gathered} z={\left[\frac{\sqrt{2}{e}^{(\pi i/2)}}{\sqrt{2}{e}^{(\pi i/4)}}\right]}^{12} \\ ={\left(e^{(\pi i/4)}\right)}^{12} \\ =\left(e^{3\pi i}\right) \end{gathered}$$

Solve further,

$z=\left[\cos \left(3\pi \right)+i\sin \left(\mathrm{\pi i}\right)\right]$

z = -1

Hence, the value of the complex number is z = -1 .

The graph z= -1 of is as follows: