Divide.
StartFraction 2 plus i Over 2 minus i EndFraction
2+i
2−i

To divide the complex numbers \(\frac{2 + i}{2 - i}\), we use the method of multiplying by the conjugate of the denominator. The conjugate of \(2 - i\) is \(2 + i\). Here's the step-by-step process:

1. **Multiply the numerator and the denominator by the conjugate of the denominator**:
   \[
   \frac{2 + i}{2 - i} \times \frac{2 + i}{2 + i} = \frac{(2 + i) \times (2 + i)}{(2 - i) \times (2 + i)}
   \]

2. **Expand the numerator and the denominator**:
   - **Numerator**: 
     \[
     (2 + i) \times (2 + i) = 2^2 + 2 \times 2 \times i + i^2 = 4 + 4i + i^2
     \]
     Since \(i^2 = -1\):
     \[
     4 + 4i + (-1) = 3 + 4i
     \]

   - **Denominator**:
     \[
     (2 - i) \times (2 + i) = 2^2 - (i)^2 = 4 - (-1) = 4 + 1 = 5
     \]

3. **Combine the results**:
   \[
   \frac{3 + 4i}{5}
   \]

4. **Separate into real and imaginary parts**:
   \[
   \frac{3 + 4i}{5} = \frac{3}{5} + \frac{4i}{5}
   \]

So, the final result is:
\[
\frac{2 + i}{2 - i} = \frac{3}{5} + \frac{4i}{5}
\]

Would you like more details or have any questions about this process?

Here are 5 related questions to expand your understanding:

1. How do you find the conjugate of a complex number?
2. What is the significance of multiplying by the conjugate when dividing complex numbers?
3. Can you explain why \(i^2 = -1\) and how it affects complex number multiplication?
4. How do you add and subtract complex numbers?
5. What is the geometric interpretation of complex number division?

**Tip:** When dividing complex numbers, multiplying by the conjugate helps eliminate the imaginary part in the denominator, simplifying the expression to a standard form.

Divide. StartFraction 2 plus i Over 2 minus i EndFraction

Learn how to divide complex numbers using the conjugate method with a step-by-step solution. In this problem, we simplify \(\frac{2 + i}{2 - i}\), focusing on key concepts like complex conjugates and imaginary unit properties (i^2 = -1). Suitable for high school students (Grades 9-12).

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