Let's analyze the complex number $w$ given by the expression:

$w = \frac{(1 - \sqrt{3}i)^5}{(1 + i)^4}$

Step 1: Express in Polar Form

Denominator: $1 + i$

To express $1 + i$ in polar form, we calculate its modulus $r$ and argument $\theta$.

• Modulus: $r = |1 + i| = \sqrt{1^2 + 1^2} = \sqrt{2}$

• Argument: $\theta = \tan^{-1}\left(\frac{1}{1}\right) = \frac{\pi}{4}$

Thus, in polar form: $1 + i = \sqrt{2} \text{cis} \frac{\pi}{4}$ where "cis" stands for $\cos \theta + i \sin \theta$.

Now raise this to the 4th power: $(1 + i)^4 = (\sqrt{2})^4 \text{cis} \left(4 \times \frac{\pi}{4}\right) = 4 \text{cis} \pi = -4$

Numerator: $1 - \sqrt{3}i$

Similarly, express $1 - \sqrt{3}i$ in polar form.

• Modulus: $r = |1 - \sqrt{3}i| = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{4} = 2$

• Argument: $\theta = \tan^{-1}\left(\frac{-\sqrt{3}}{1}\right) = -\frac{\pi}{3}$

Thus, in polar form: $1 - \sqrt{3}i = 2 \text{cis} \left(-\frac{\pi}{3}\right)$

Now raise this to the 5th power: $(1 - \sqrt{3}i)^5 = 2^5 \text{cis} \left(5 \times \left(-\frac{\pi}{3}\right)\right) = 32 \text{cis} \left(-\frac{5\pi}{3}\right)$

Step 2: Combine to Find $w$

Now, combine the numerator and denominator: