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To solve the expression \(\frac{8 - 7i}{8 + i}\), we need to simplify it by eliminating the imaginary part in the denominator. We do this by multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(8 + i\) is \(8 - i\).

Here's the process:

1. **Multiply both the numerator and the denominator by the conjugate of the denominator:**
   \[
   \frac{8 - 7i}{8 + i} \cdot \frac{8 - i}{8 - i} = \frac{(8 - 7i)(8 - i)}{(8 + i)(8 - i)}
   \]

2. **Expand the numerator:**
   \[
   (8 - 7i)(8 - i) = 8 \cdot 8 - 8 \cdot i - 7i \cdot 8 + 7i \cdot i
   \]
   Simplifying each term:
   \[
   = 64 - 8i - 56i + 7i^2
   \]
   Since \(i^2 = -1\), substitute \(7i^2 = 7(-1) = -7\):
   \[
   = 64 - 8i - 56i - 7 = 57 - 64i
   \]

3. **Expand the denominator:**
   \[
   (8 + i)(8 - i) = 8^2 - i^2 = 64 - (-1) = 64 + 1 = 65
   \]

4. **Combine the results:**
   \[
   \frac{8 - 7i}{8 + i} = \frac{57 - 64i}{65}
   \]

5. **Separate real and imaginary parts:**
   \[
   = \frac{57}{65} - \frac{64}{65}i
   \]

So, the simplified answer in the form \(a + bi\) is:
\[
\frac{57}{65} - \frac{64}{65}i
\]

Would you like further clarification on any step?

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Here are five related questions for further practice:

1. How do you multiply complex numbers?
2. Why do we use the conjugate when simplifying complex fractions?
3. What is the significance of \(i^2 = -1\) in complex number calculations?
4. How do you convert a complex number to polar form?
5. Can you add or subtract complex numbers in the same way as real numbers?

**Tip:** When dealing with complex fractions, always check if you can simplify by multiplying by the conjugate.

Simplify \( \frac{8 - 7i}{8 + i} \) and express the answer in the form \(a + bi\).

Learn how to simplify the complex fraction \( \frac{8 - 7i}{8 + i} \) by multiplying by the conjugate of the denominator. This solution covers essential complex number operations, such as multiplying by conjugates, and is ideal for students in Grades 10-12.

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