Question

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

role="math" localid="1646666471780" ${\left[2\left(\cos \frac{\mathrm{\pi }}{10}+i\sin \frac{\mathrm{\pi }}{10}\right)\right]}^5$

Short answer

The expression in the standard form is $32i$.

Step-by-step solution

Step 1. Given information.

Consider the given question,

${\left[2\left(\cos \frac{\mathrm{\pi }}{10}+i\sin \frac{\mathrm{\pi }}{10}\right)\right]}^5$

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 $z=\left(a+bi\right)$,

$x^5={\left[2\left(\cos \frac{\mathrm{\pi }}{10}+i\sin \frac{\mathrm{\pi }}{10}\right)\right]}^5$

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

Using the equation (i),

$$\begin{gathered} x^5=2^5\left[\cos \left(5·\frac{\mathrm{\pi }}{10}\right)+i\sin \left(5·\frac{\mathrm{\pi }}{10}\right)\right] \\ {x}^5=32\left(\cos \frac{\mathrm{\pi }}{2}+i\sin \frac{\mathrm{\pi }}{2}\right) \\ {x}^5=32\left(0+i\right) \\ {x}^5=32i \end{gathered}$$