Q32P - 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.

 ${\left(2-3i\right)}^{\mathbf{4}}$.

Step-by-Step Solution

Answer

The absolute value is $\left|z\right|=169$ .

Step 1: Given Information

The complex number is ${\left(2-3i\right)}^4$.

Step 2: Absolute value of the Complex number

Let z be a complex number consisting of two real numbers as:
$\mathit{z}\mathbf{=}\mathit{x}\mathbf{+}{}_{\mathbf{z}}\mathit{y}$

Here x and y are the real numbers.

 

The absolute value of z can be calculated as:

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

Step 3: Find the value

The complex number is ${\left(2-3i\right)}^4$ .

 

The absolute value is given below.

$$\begin{gathered} \left|z\right|=\left|{\left(2-3i\right)}^4\right| \\ ={\left|2-3i\right|}^4 \\ ={\left(\sqrt{4+9}\right)}^4 \\ \left|z\right|=169 \end{gathered}$$

 

Thus, the final answer is obtained as:

$\left|z\right|=169$ .