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.

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

Short answer

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

Step-by-step solution

Given Information

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

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:

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

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:

$\left|z\right|=5\sqrt{5}$.