Q34P - Mathematical Methods in Physical Sciences

Question

Find the absolute value of each of the following using the discussion above. Try to do simple problems like these in your head-it saves time.

${\mathbf{\left(}\frac{\mathbf{1}\mathbf{+}\mathbf{i}}{\mathbf{1}\mathbf{-}\mathbf{i}}\mathbf{\right)}}^{\mathbf{5}}$

Step-by-Step Solution

Answer

The answer is obtained ${\left|\frac{1+i}{1-i}\right|}^5=1$

Step 1: Given information

It is given that $z={\left|\frac{1+i}{1-i}\right|}^5$

Step 2: Absolute value of the Complex number

Let z be a complex number consisting of two real numbers as:

 $z=x+ty$

Here x and y are the real numbers.

 

The absolute value of z can be calculated as:

$\left|\mathrm{z}\right|=\sqrt{{\mathrm{x}}^2+{\mathrm{y}}^2}$

Step 3: Begin by stating the given information.

State the given information.

$z={\left(\frac{1+i}{1-i}\right)}^5$

Step 4: Find the modulus of the function.

Find the modulus of the function.

$$\begin{gathered} \left|z\right|=\left|{\left(\frac{1+i}{1-i}\right)}^5\right| \\ {\left|\frac{1+i}{1-i}\right|}^5 \end{gathered}$$

 

The absolute value can be calculated as below:

 $$\begin{gathered} {\left|\frac{1+i}{1-i}\right|}^5={\left(\frac{\sqrt{1+1}}{\sqrt{1+1}}\right)}^5 \\ ={\left(\frac{\sqrt{2}}{\sqrt{2}}\right)}^5 \\ =1 \end{gathered}$$

 

Thus, the final answer is obtained as:

${\left|\frac{1+i}{1-i}\right|}^5=1$