The expression you've given is \( w = -8e^{i \arctan\left(-\sqrt{\frac{7}{3}}\right)} \).

This expression represents a complex number in polar form, where the magnitude is \(8\) and the argument (angle) is given by \( \arctan\left(-\sqrt{\frac{7}{3}}\right) \). Let's break down the components:

1. **Magnitude**: The magnitude (or modulus) of \( w \) is \( |w| = 8 \).

2. **Argument**: The argument (or angle) \( \theta \) is \( \theta = \arctan\left(-\sqrt{\frac{7}{3}}\right) \).

3. **Complex Form**: In rectangular (Cartesian) form, the complex number \( w \) can be expressed as:
   \[
   w = -8 \left( \cos\theta + i\sin\theta \right)
   \]
   where \( \theta = \arctan\left(-\sqrt{\frac{7}{3}}\right) \).

To simplify or convert this into standard rectangular form, let's compute \( \theta \).

### Step 1: Calculate \( \theta \)
\[
\theta = \arctan\left(-\sqrt{\frac{7}{3}}\right)
\]

### Step 2: Convert to Rectangular Form
Using Euler's formula \( e^{i\theta} = \cos\theta + i\sin\theta \), we get:
\[
w = -8 \left(\cos(\theta) + i\sin(\theta)\right)
\]

Let's calculate the cosine and sine of \( \theta \).

### Step 3: Calculation
1. Calculate \( \theta \).
2. Determine the cosine and sine values for \( \theta \).
3. Substitute these values into the equation for \( w \).

I'll compute these values next.The complex number \( w \) in rectangular form is approximately:

\[
w \approx -4.38 + 6.69i
\]

Here, the real part is \(-4.38\) and the imaginary part is \(6.69\).

Would you like more details on the process, or do you have any specific questions?

### Related Questions:
1. How do you convert a complex number from polar to rectangular form?
2. What is the geometric interpretation of a complex number in polar form?
3. How do you calculate the argument of a complex number?
4. What is Euler's formula, and how is it used in complex numbers?
5. How can you find the magnitude of a complex number in rectangular form?
6. How do the trigonometric identities relate to complex numbers?
7. What are the applications of complex numbers in engineering and physics?
8. Can you describe the process to manually calculate the cosine and sine of an angle?

**Tip**: Understanding the relationship between the polar and rectangular forms of complex numbers can help you visualize and manipulate them more effectively.

Explore the complex number -8e^{i \arctan(-\sqrt{\frac{7}{3}})}, its representation in polar form, and conversion to rectangular form. Learn about Euler's formula and its application in computing complex numbers. Suitable for advanced high school students and above.

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