Question

Write the expression in the standard form $a+ib$.

${\left[\frac{1}{2}\left(\cos 72°+i\sin 72°\right)\right]}^5$

Short answer

The expression in the standard form is $\frac{1}{32}$.

Step-by-step solution

Step 1. Given information.

Consider the given question,

${\left[\frac{1}{2}\left(\cos 72°+i\sin 72°\right)\right]}^5$

We know that for a complex number raise to n when it is in the polar form $z^n=r^n\left(\cos \theta +i\sin \left(n\theta \right)\right)......\left(i\right)$

Write the expression in the standard form $z=\left(a+bi\right)$,

$z^5={\left[\frac{1}{2}\left(\cos 72°+i\sin 72°\right)\right]}^5$

Step 2. Use equation (i) and solve the equation.

Using the equation (i),

$$\begin{gathered} z^5={\left(\frac{1}{2}\right)}^5\left[\cos \left(5·72°\right)+i\sin \left(5·72°\right)\right] \\ {z}^5=\frac{1}{32}\left(\cos 360°+i\sin 360°\right) \\ {z}^5=\frac{1}{32}\left(1+i·0\right) \\ {z}^5=\frac{1}{32} \end{gathered}$$