https://file.aiquickdraw.com/mathbot/file/1725798560419ap7mp846.jpg

It seems there was an issue processing the image. I will manually transcribe the mathematical expression and solve the problem. The expression shown is:

\[
z = \frac{(1 + i)^4 \cdot (1 + i\sqrt{3})^5}{(2 \cos \frac{\pi}{20} + 2 i \sin \frac{\pi}{20})^6}
\]

Now, let's solve this complex expression step by step.

### Step 1: Simplify the numerator
We'll begin by simplifying the terms \((1 + i)^4\) and \((1 + i\sqrt{3})^5\). We can use polar forms for these complex numbers.

#### Simplifying \((1 + i)^4\)
The complex number \(1 + i\) can be written in polar form as:
\[
1 + i = \sqrt{2} \text{cis} \frac{\pi}{4}
\]
So,
\[
(1 + i)^4 = (\sqrt{2})^4 \text{cis} \left(4 \cdot \frac{\pi}{4}\right) = 16 \text{cis} \pi = -16
\]

#### Simplifying \((1 + i\sqrt{3})^5\)
The complex number \(1 + i\sqrt{3}\) can be written in polar form as:
\[
1 + i\sqrt{3} = 2 \text{cis} \frac{\pi}{3}
\]
So,
\[
(1 + i\sqrt{3})^5 = 2^5 \text{cis} \left(5 \cdot \frac{\pi}{3}\right) = 32 \text{cis} \frac{5\pi}{3}
\]

### Step 2: Simplify the denominator
The denominator contains the expression \(\left(2 \cos \frac{\pi}{20} + 2 i \sin \frac{\pi}{20}\right)\). This can be rewritten using Euler's formula:
\[
2 \cos \frac{\pi}{20} + 2 i \sin \frac{\pi}{20} = 2 \text{cis} \frac{\pi}{20}
\]
Thus,
\[
\left(2 \cos \frac{\pi}{20} + 2 i \sin \frac{\pi}{20}\right)^6 = 2^6 \text{cis} \left(6 \cdot \frac{\pi}{20}\right) = 64 \text{cis} \frac{3\pi}{10}
\]

### Step 3: Combine the results
Now we combine the simplified terms from the numerator and denominator:
\[
z = \frac{-16 \cdot 32 \text{cis} \frac{5\pi}{3}}{64 \text{cis} \frac{3\pi}{10}} = \frac{-512 \text{cis} \frac{5\pi}{3}}{64 \text{cis} \frac{3\pi}{10}} = -8 \text{cis} \left(\frac{5\pi}{3} - \frac{3\pi}{10}\right)
\]
We now calculate the angle difference:
\[
\frac{5\pi}{3} - \frac{3\pi}{10} = \frac{50\pi}{30} - \frac{9\pi}{30} = \frac{41\pi}{30}
\]

Thus,
\[
z = -8 \text{cis} \frac{41\pi}{30}
\]

### Final Answer
The solution in polar form is:
\[
z = -8 \text{cis} \frac{41\pi}{30}
\]

Let me know if you want more details or have any questions.

Here are some related questions:
1. How do you convert a complex number into polar form?
2. What is Euler's formula and how is it applied to complex numbers?
3. How do you multiply complex numbers in polar form?
4. What is the significance of the argument (angle) in polar form?
5. Can this solution be expressed in rectangular form?

**Tip:** When dealing with powers of complex numbers, converting to polar form can make calculations involving powers and products much simpler.

Solve the complex expression: z = ((1 + i)^4 * (1 + i√3)^5) / (2cos(π/20) + 2isin(π/20))^6

Explore the solution to a complex numbers problem involving powers, polar form, and trigonometric simplifications. This problem showcases the use of Euler's formula and De Moivre's theorem to simplify and calculate powers of complex numbers. Suitable for undergraduate mathematics students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation