Express each of the following complex numbers in polar form, and plot them in the complex plane, indicating the magnitude and angle of each number.

A) $1 + j\sqrt{3}$

$r = \sqrt{1^2 + \sqrt{3}^2} = \sqrt{4} = 2$

$\theta = arctan(\sqrt{3}/1) = arctan(\sqrt{3}) = \frac{\pi}{3}$

Therefore the polar form of this complex number is: $2e^{j\frac{\pi}{3}}$

B) $-5$

$r = 5$

$\theta = \pi$

Therefore the polar form of this complex number is: $5e^{j\pi}$

F) $(1 + j)^{5}$

$r = \sqrt{1^2 + 1^2} = \sqrt{2}$

$\theta = \frac{\pi}{4}$

$(1 + j) = \sqrt{2}e^{j\frac{\pi}{4}}$

$(1 + j)^{5} = (\sqrt{2}e^{j\frac{\pi}{4}})^{5} = 2^{\frac{5}{2}}e^{j\frac{5\pi}{4}} = 4\sqrt{2}e^{j(\pi + \frac{\pi}{4})} =4\sqrt{2}e^{j\pi}e^{j\frac{\pi}{4}} = -4(\sqrt{2}e^{j\frac{\pi}{4}}) = -4(1 + j)$

Therefore the polar form of this complex number is: $-4(\sqrt{2}e^{j\frac{\pi}{4}})$

I) $\frac{1 + j\sqrt{3}}{\sqrt{3} + j}$

$r = 2$

$Equation 1 = 1 + j\sqrt{3} => \theta_{1} = \frac{\pi}{3}$

$Equation 2 = \sqrt{3} + j => \theta_{2} = \frac{\pi}{6}$

$\frac{2e^{j\frac{\pi}{3}}}{2e^{j\frac{\pi}{6}}} = \frac{e^{j\frac{\pi}{3}}}{e^{j\frac{\pi}{6}}} = e^{j(\frac{\pi}{3} - \frac{\pi}{6})} = e^{j\frac{\pi}{6}}$

Therefore the polar form of this complex number is: $e^{j\frac{\pi}{6}}$