Let 
v
=
−
1
−
i
v=−1−i and 
w
=
−
3
−
3
3
i
w=−3−3 
3
​
 i.



Determine modulus 
r
r and argument 
θ
θ of 
z
=
v
3
w
2
z= 
w 
2
 
v 
3
 
​
 .

r
=
r=

 
 
 
 
 
 


To solve for the modulus \( r \) and argument \( \theta \) of the complex number \( z = \frac{v^3}{w^2} \), we will follow these steps:

### Step 1: Express \( v \) and \( w \) in standard form
Given:
\[
v = -1 - i
\]
\[
w = -3 - 3\sqrt{3}i
\]

### Step 2: Compute the modulus and argument of \( v \) and \( w \)
1. **Modulus of \( v \):**
   \[
   |v| = \sqrt{(-1)^2 + (-1)^2} = \sqrt{2}
   \]
2. **Argument of \( v \):**
   \[
   \theta_v = \text{arg}(-1 - i) = \tan^{-1}\left(\frac{-1}{-1}\right) = \tan^{-1}(1) = -\frac{3\pi}{4}
   \]
   (since \( v \) lies in the third quadrant).

3. **Modulus of \( w \):**
   \[
   |w| = \sqrt{(-3)^2 + (-3\sqrt{3})^2} = \sqrt{9 + 27} = \sqrt{36} = 6
   \]
4. **Argument of \( w \):**
   \[
   \theta_w = \text{arg}(-3 - 3\sqrt{3}i) = \tan^{-1}\left(\frac{-3\sqrt{3}}{-3}\right) = \tan^{-1}(\sqrt{3}) = -\frac{2\pi}{3}
   \]
   (since \( w \) lies in the third quadrant).

### Step 3: Compute the modulus and argument of \( z \)
Using the properties of modulus and arguments of complex numbers:

1. **Modulus of \( z \):**
   \[
   |z| = \left|\frac{v^3}{w^2}\right| = \frac{|v|^3}{|w|^2} = \frac{(\sqrt{2})^3}{6^2} = \frac{2\sqrt{2}}{36} = \frac{\sqrt{2}}{18}
   \]
2. **Argument of \( z \):**
   \[
   \theta_z = 3\theta_v - 2\theta_w = 3\left(-\frac{3\pi}{4}\right) - 2\left(-\frac{2\pi}{3}\right) = -\frac{9\pi}{4} + \frac{4\pi}{3} = -\frac{27\pi + 16\pi}{12} = -\frac{11\pi}{12}
   \]

Thus, the modulus and argument of \( z \) are:
\[
r = \frac{\sqrt{2}}{18}, \quad \theta = -\frac{11\pi}{12}
\]

Would you like further details or have any questions?

---

Here are 5 questions to deepen your understanding:
1. How do you calculate the modulus of a complex number?
2. Why does the argument depend on which quadrant the complex number lies in?
3. Can the argument of a complex number exceed \( 2\pi \) or go below \( 0 \)?
4. How does raising a complex number to a power affect its modulus and argument?
5. What happens to the argument of a quotient of two complex numbers?

**Tip:** Always express angles for arguments in radians unless specifically asked for degrees, as radians are standard in most complex number problems.

Let v = -1 - i and w = -3 - 3√3i. Determine modulus r and argument θ of z = v^3 / w^2.

Learn how to calculate the modulus and argument of the complex number z = v^3 / w^2, where v = -1 - i and w = -3 - 3√3i. This solution involves key concepts such as the modulus and argument of complex numbers, De Moivre's Theorem, and polar form representation.

Math Problem Statement

Let v

− 1 − i v=−1−i and w

Determine modulus r r and argument θ θ of z

r

Solution

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