To simplify the expression \(\frac{7 + i}{7 - i}\), we can multiply the numerator and the denominator by the conjugate of the denominator to remove the imaginary part from the denominator.

Here's the step-by-step process:

1. **Identify the conjugate of the denominator:**

   The conjugate of \(7 - i\) is \(7 + i\).

2. **Multiply the numerator and the denominator by the conjugate:**

   \[
   \frac{7 + i}{7 - i} \times \frac{7 + i}{7 + i} = \frac{(7 + i)(7 + i)}{(7 - i)(7 + i)}
   \]

3. **Simplify the denominator using the difference of squares formula:**

   \[
   (7 - i)(7 + i) = 7^2 - i^2 = 49 - (-1) = 49 + 1 = 50
   \]

4. **Simplify the numerator by expanding it:**

   \[
   (7 + i)(7 + i) = 7 \times 7 + 7 \times i + 7 \times i + i \times i = 49 + 14i + i^2 = 49 + 14i - 1 = 48 + 14i
   \]

5. **Combine and simplify:**

   \[
   \frac{48 + 14i}{50} = \frac{48}{50} + \frac{14i}{50} = \frac{24}{25} + \frac{7}{25}i
   \]

So, the simplified form of \(\frac{7 + i}{7 - i}\) is:

\[
\frac{24}{25} + \frac{7}{25}i
\]

Would you like more details or have any questions?

Here are some related questions for further exploration:

1. How do you multiply two complex numbers?
2. What is the conjugate of a complex number and why is it useful?
3. How can you rationalize a complex number in the denominator?
4. How do you add and subtract complex numbers?
5. What is the geometric interpretation of multiplying complex numbers?

**Tip:** When dividing complex numbers, using the conjugate of the denominator helps eliminate the imaginary part, making the expression easier to handle in standard form.

Learn how to simplify the division of complex numbers (7 + i) / (7 - i) using the conjugate and difference of squares formula. This problem explores key concepts in complex numbers and algebraic manipulation. Suitable for advanced high school students.

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