Q23P - Mathematical Methods in Physical Sciences

Question

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.

23.  $\frac{{(1+i)}^{48}}{{(\sqrt{3}-i)}^{25}}$

Step-by-Step Solution

Answer

The value of the expression  $\frac{{(1+i)}^{48}}{{(\sqrt{3}-i)}^{25}}$ is $\frac{(\sqrt{3}+i)}{4}$ .

 

The graph of $\frac{(\sqrt{3}+i)}{4}$  is shown below.

 

Step 1: Given information

The given complex number is $\frac{{(1+i)}^{48}}{{(\sqrt{3}-i)}^{25}}$ .

Step 2: Definition of complex numbers

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.

Step 3: Find the value of z 1

Let the given number be     $z=\frac{{(1+i)}^{48}}{{(\sqrt{3}-i)}^{25}}$                   … (1)

 

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

 

Let $z_1=1+i$ .

 

The modulus of $z_1$ is $r_1=\sqrt{1+1}$ .

 

That is  $r_1=\sqrt{2}$.

 

The argument of $z_1$ is ${\theta }_1=\arctan \left(1\right)$ .

 

That is ${\theta }_1=\frac{\pi }{4}$ .

 

Hence  $z_1=\sqrt{2}{e}^{(\pi i/4)}$.                        … (2)

 

Step 4: Find the value of z 2

Let  $z_2=\sqrt{3}-i$.

 

The modulus of ${z_2}_{}$is $r_2=\sqrt{3+1}$ .

 

That is  $r_1=2$.

 

The argument of  $z_2$ is ${\theta }_2=-\arctan \left(\frac{1}{\sqrt{3}}\right)$  .

 

That is ${\theta }_2=-\frac{\pi }{6}$  .

 

The angle is in fourth quadratic. Add $2\pi$  to ${\theta }_2$  .

 

 $\begin{array}{l}{\theta }_2=2\pi -\frac{\pi }{6}\\ {\theta }_2=\frac{11}{6}\pi \end{array}$

 

Hence,  $z_2=2e^{(11\pi i/4)}$.                                                      … (3)

 

Step 5: Find the value of the complex number z

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

 $\begin{array}{c}z=\frac{{\left[\sqrt{2}{e}^{(\pi i/4)}\right]}^{48}}{{\left[2e^{(11\pi i/6)}\right]}^{25}}\\ =\frac{{\left[\sqrt{2}{e}^{(\pi i/4)}\right]}^{48}}{{\left[2e^{(11\pi i/6)}\right]}^{24}\left[2e^{(11\pi i/6)}\right]}\\ =\frac{{\left(2\right)}^{24}{e}^{\left(12\pi i\right)}}{{\left(2\right)}^{24}{e}^{\left(44\pi i\right)}}\times \frac{1}{\left[2e^{(11\pi i/6)}\right]}\\ =\frac{\cos \left(12\pi \right)+i\sin \left(12\pi \right)}{\cos \left(44\pi \right)+i\sin \left(44\pi \right)}\times \frac{1}{2}{e}^{(-11\pi i/6)}\end{array}$

 

Solve further,

 $\begin{array}{c}z=\frac{1}{2}\left[\cos \left(\frac{-11\pi }{6}\right)+i\sin \left(\frac{-11\pi }{6}\right)\right]\\ =\frac{1}{2}\left[\frac{\sqrt{3}}{2}+i\frac{1}{2}\right]\\ z=\frac{(\sqrt{3}+i)}{4}\end{array}$

Hence, the value of the complex number is $z=\frac{(\sqrt{3}+i)}{4}$ .

Step 6: Plot the complex number

The graph of  $z=\frac{(\sqrt{3}+i)}{4}$ as follows: