Q29P - 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.

 ${\left(1+2\mathit{i}\right)}^{\mathbf{3}}$.

Step-by-Step Solution

Answer

The absolute value is $\left|z\right|=5\sqrt{5}$ .

Step 1: Given Information

The complex number is ${\left(1=2i\right)}^3$.

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:

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

Step 3: Find the value

The complex number is ${\left(1+2i\right)}^3$ .

 

The absolute value can be calculated as below:

$$\begin{gathered} \left|z\right|={\left(1=2i\right)}^3 \\ ={\sqrt{1+4}}^3 \\ =\sqrt{5}\times \sqrt{5}\times \sqrt{5} \\ \left| z\right|=5\sqrt{5} \end{gathered}$$

 

Thus, the final answer is obtained as:

$\left|z\right|=5\sqrt{5}$ .