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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:

\[
w = \frac{32 \text{cis} \left(-\frac{5\pi}{3}\right)}{-4}
= -8 \text{cis} \left(-\frac{5\pi}{3} - \pi\right)
= -8 \text{cis} \left(-\frac{8\pi}{3}\right)
\]

Since the argument \( -\frac{8\pi}{3} \) is coterminal with \( \frac{4\pi}{3} \), we have:
\[
w = -8 \text{cis} \frac{4\pi}{3}
\]

Thus, the polar form of \( w \) is:
\[
w = 8 \text{cis} \left( \frac{4\pi}{3} \right)
\]

### Step 3: Convert to Cartesian Form
To convert back to Cartesian form, use the definition of "cis":

\[
w = 8 \left( \cos \frac{4\pi}{3} + i \sin \frac{4\pi}{3} \right)
\]

We know:
\[
\cos \frac{4\pi}{3} = -\frac{1}{2}, \quad \sin \frac{4\pi}{3} = -\frac{\sqrt{3}}{2}
\]

Thus:
\[
w = 8 \left( -\frac{1}{2} - i \frac{\sqrt{3}}{2} \right) = -4 - 4\sqrt{3}i
\]

### Final Answer:
- **Polar form:** \( w = 8 \text{cis} \left( \frac{4\pi}{3} \right) \)
- **Cartesian form:** \( w = -4 - 4\sqrt{3}i \)

Would you like further details or have any specific questions?

Here are 5 questions related to this problem:
1. How do you calculate the modulus and argument of a complex number?
2. Why do we use polar form to simplify powers of complex numbers?
3. What does "cis" mean in polar form?
4. How do arguments change when raising a complex number to a power?
5. Why is the argument in the final polar form expressed as \( \frac{4\pi}{3} \)?

**Tip:** When converting between Cartesian and polar forms, always verify your calculations of the modulus and argument to ensure accuracy.

Consider the following complex number: w = (1 - √3i)^5 / (1 + i)^4. (a) Express w in Cartesian form and in polar form.

Learn how to express the complex number w = (1 - √3i)^5 / (1 + i)^4 in both Cartesian and polar forms. The solution covers key concepts of complex numbers, including finding modulus, argument, and using De Moivre's Theorem for simplification. Suitable for undergraduate mathematics students.

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