Question

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

${\left[4\left(\cos 40°+i\sin 40°\right)\right]}^3$

Short answer

The expression in the standard form is $-32+32\sqrt{3}i$.

Step-by-step solution

Step 1. Given information.

Consider the given question,

${\left[4\left(\cos 40°+i\sin 40°\right)\right]}^3$

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

$z^3={\left[4\left(\cos 40°+i\sin 40°\right)\right]}^3$

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

Using the equation (i),

$$\begin{gathered} z^3=4^3\left[\cos \left(3·40°\right)+i\sin \left(3·40°\right)\right] \\ {z}^3=64\left[\cos 120°+i\sin 120°\right] \\ {z}^3=64\left(-\frac{1}{2}+i\frac{\sqrt{3}}{2}\right) \\ {z}^3=-32+i\left(32·\sqrt{3}\right) \end{gathered}$$