Question

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

${\left[5\left(\cos \frac{3\mathrm{\pi }}{16}+i\sin \frac{3\mathrm{\pi }}{16}\right)\right]}^4$

Short answer

The expression in the standard form is $-\frac{25\sqrt{2}}{2}+\frac{25\sqrt{2}}{2}i$.

Step-by-step solution

Step 1. Given information.

Consider the given question,

${\left[5\left(\cos \frac{3\mathrm{\pi }}{16}+i\sin \frac{3\mathrm{\pi }}{16}\right)\right]}^4$

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

$z^4={\left[\sqrt{5}\left(\cos \frac{3\mathrm{\pi }}{16}+i\sin \frac{3\mathrm{\pi }}{16}\right)\right]}^4$

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

Using the equation (i),

$$\begin{gathered} z^4={\left(\sqrt{5}\right)}^4\left[\cos \left(4·\frac{3\mathrm{\pi }}{16}\right)+i\sin \left(4·\frac{3\mathrm{\pi }}{16}\right)\right] \\ {z}^4=5^2\left(\cos \frac{3\mathrm{\pi }}{4}+i\sin \frac{3\mathrm{\pi }}{4}\right) \\ {z}^4=25\left(-\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i\right) \\ {z}^4=-\frac{25\sqrt{2}}{2}+\frac{25\sqrt{2}}{2}i \end{gathered}$$