https://file.aiquickdraw.com/mathbot/file/1722754950006xvgjx8ue.jpg

solve question 11 Z1 /Z2 by taking conjugate

To solve question 11:

### Given:
- \( z_1 = 5 + 3i \)
- \( z_2 = 4 - 2\sqrt{-1} \)

### Find:
\[ \frac{z_1}{z_2} \]

### Step-by-Step Solution:

1. **Write the given complex numbers**:
   \[ z_1 = 5 + 3i \]
   \[ z_2 = 4 - 2i \] (since \(\sqrt{-1} = i\))

2. **Multiply the numerator and denominator by the conjugate of the denominator**:
   \[ \frac{z_1}{z_2} = \frac{5 + 3i}{4 - 2i} \]
   Conjugate of \( z_2 = 4 - 2i \) is \( 4 + 2i \).

3. **Apply the multiplication**:
   \[ \frac{5 + 3i}{4 - 2i} \cdot \frac{4 + 2i}{4 + 2i} = \frac{(5 + 3i)(4 + 2i)}{(4 - 2i)(4 + 2i)} \]

4. **Simplify the denominator**:
   \[ (4 - 2i)(4 + 2i) = 4^2 - (2i)^2 = 16 - 4(-1) = 16 + 4 = 20 \]

5. **Simplify the numerator**:
   \[ (5 + 3i)(4 + 2i) = 5 \cdot 4 + 5 \cdot 2i + 3i \cdot 4 + 3i \cdot 2i \]
   \[ = 20 + 10i + 12i + 6i^2 \]
   \[ = 20 + 22i + 6(-1) \]
   \[ = 20 + 22i - 6 \]
   \[ = 14 + 22i \]

6. **Combine the results**:
   \[ \frac{14 + 22i}{20} = \frac{14}{20} + \frac{22i}{20} \]
   \[ = \frac{7}{10} + \frac{11i}{10} \]

### Final Answer:
\[ \frac{z_1}{z_2} = \frac{7}{10} + \frac{11i}{10} \]

Would you like details or have any questions on the solution?

### Related Questions:
1. What is the general form of a complex number?
2. How do you find the conjugate of a complex number?
3. Why do we use the conjugate to divide complex numbers?
4. What is the geometric interpretation of multiplying by a complex conjugate?
5. How do you simplify the expression involving \(i^2\)?
6. Can you solve \(\frac{3 + 4i}{1 - 2i}\)?
7. How do you add and subtract complex numbers?
8. What are the applications of complex numbers in real-world problems?

### Tip:
When dividing complex numbers, always multiply the numerator and the denominator by the conjugate of the denominator to simplify the expression.

Learn how to solve question 11 Z1 /Z2 using conjugates in complex number division. This problem involves finding the division of complex numbers, understanding conjugates, and applying basic algebraic operations with complex numbers. Suitable for high school 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