Answer
$-\frac{1 }{2} +i \frac{\sqrt 3 }{2}$
Work Step by Step
Step 1: Simplify using exponent rule $(a^n)^m=a^{n\times m}$
$\left(e^{\frac{i\pi}{3}}\right)^2=e^{\frac{2\pi i}{3}}$
Step 2: Write polar form $z=re^{i\theta}$ to find values of $r$ and $\theta$:
$z=re^{i\theta}=1e^{\frac{2\pi i}{3}} \Rightarrow ~~ r=1 ~,~\theta=\frac{2\pi }{3}$
Step 3: Use Euler formula: $z=re^{i\theta}= r(\cos \theta +i \sin \theta)$ to write in Cartesian form
$$1e^{\frac{2\pi i}{3}}= 1\left(\cos \frac{2\pi }{3} +i \sin \frac{2\pi }{3}\right) $$ Simplify
$$\left(e^{\frac{\pi i}{3}}\right)^2= \frac{-1 }{2} +i \frac{\sqrt 3 }{2} $$
Step 4: The Cartesian form of $\left(e^{\frac{i\pi}{3}}\right)^2$ is $-\frac{1 }{2} +i \frac{\sqrt 3 }{2}$.
