Let z*z* = −1+i−1+*i* and w*w* = −1+3^0,5i−1+3^0,5​*i*.

Determine the modulus r*r* of z3w5*z*3*w*5.     

r=*r*=Determine the argument θ*θ* of z3w5*z*3*w*5. 


Choose θ*θ* such that −π< θ≤π−*π*< *θ*≤*π*.

To solve this problem, we need to determine both the modulus \( r \) and the argument \( \theta \) of the complex expression \( z^3 w^5 \).

### Step 1: Simplifying the expression for \( z^3 w^5 \)
Given:
- \( z \cdot \bar{z} = -1 + i \)
- \( w \cdot \bar{w} = -1 + \sqrt{3} i \)

We need to first express \( z \) and \( w \) in polar form to make it easier to compute powers and products.

### Step 2: Finding \( z \)
The relation \( z \cdot \bar{z} = -1 + i \) tells us that \( z \cdot \bar{z} \) is the modulus squared of \( z \), which is equal to:
\[
|z|^2 = \sqrt{(-1)^2 + 1^2} = \sqrt{2}
\]
Therefore, the modulus of \( z \) is \( |z| = \sqrt[4]{2} \).

Next, to determine the argument of \( z \), we take the argument of the complex number \( -1 + i \). The argument \( \theta_z \) of \( -1 + i \) is:
\[
\theta_z = \text{arg}(-1 + i) = \tan^{-1} \left( \frac{1}{-1} \right) = \frac{3\pi}{4}
\]

Thus, we can express \( z \) as:
\[
z = \sqrt[4]{2} e^{i \frac{3\pi}{4}}
\]

### Step 3: Finding \( w \)
For \( w \), the relation \( w \cdot \bar{w} = -1 + \sqrt{3} i \) implies that the modulus squared of \( w \) is:
\[
|w|^2 = \sqrt{(-1)^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = 2
\]
So the modulus of \( w \) is \( |w| = \sqrt[4]{2} \).

To find the argument of \( w \), we use the argument of the complex number \( -1 + \sqrt{3} i \). This gives:
\[
\theta_w = \text{arg}(-1 + \sqrt{3} i) = \tan^{-1} \left( \frac{\sqrt{3}}{-1} \right) = \frac{2\pi}{3}
\]

Thus, we can express \( w \) as:
\[
w = \sqrt[4]{2} e^{i \frac{2\pi}{3}}
\]

### Step 4: Calculating \( z^3 w^5 \)
Now we compute the modulus and argument of \( z^3 w^5 \). Using the property of exponentiation in polar form:

1. **Modulus**:
\[
|z^3 w^5| = |z|^3 \cdot |w|^5 = (\sqrt[4]{2})^3 \cdot (\sqrt[4]{2})^5 = 2^{\frac{3}{4}} \cdot 2^{\frac{5}{4}} = 2^{2} = 4
\]

2. **Argument**:
\[
\text{arg}(z^3 w^5) = 3 \cdot \frac{3\pi}{4} + 5 \cdot \frac{2\pi}{3} = \frac{9\pi}{4} + \frac{10\pi}{3} = \frac{27\pi}{12} + \frac{40\pi}{12} = \frac{67\pi}{12}
\]
We reduce the argument by subtracting \( 2\pi \) until it lies within \( (-\pi, \pi] \). Subtracting \( 2\pi \) (or \( \frac{24\pi}{12} \)) multiple times:
\[
\frac{67\pi}{12} - 2\pi \times 5 = \frac{67\pi}{12} - \frac{60\pi}{12} = \frac{7\pi}{12}
\]
Thus, the argument is \( \theta = \frac{7\pi}{12} \).

### Final Answer:
- \( r = 4 \)
- \( \theta = \frac{7\pi}{12} \)

### Would you like more details or have any questions?

Here are 5 relative questions:
1. How can we convert a complex number into polar form?
2. What is the general method to calculate the modulus of a product of complex numbers?
3. How do you find the argument of a complex number in the complex plane?
4. What steps are involved in reducing an argument to be within \( (-\pi, \pi] \)?
5. Why is polar form useful in calculating powers of complex numbers?

**Tip:** When multiplying complex numbers, converting them into polar form makes it easier to handle both modulus and argument calculations.

Let z*z* = −1+i and w*w* = −1+√3i. Determine the modulus r*r* of z^3 w^5. r=Determine the argument θ*θ* of z^3 w^5. Choose θ*θ* such that −π < θ ≤ π.

Learn how to calculate the modulus and argument of the complex expression z^3 w^5, where z = −1 + i and w = −1 + √3i. This solution uses polar form, modulus, and argument concepts, along with De Moivre's Theorem, making it suitable for undergraduate-level mathematics.

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