To simplify the expression \(\frac{5 + 7i}{1 + i}\), follow these steps:

1. **Multiply the numerator and the denominator by the conjugate of the denominator**:  
   The conjugate of \(1 + i\) is \(1 - i\).

   \[
   \frac{5 + 7i}{1 + i} \times \frac{1 - i}{1 - i} = \frac{(5 + 7i)(1 - i)}{(1 + i)(1 - i)}
   \]

2. **Expand the numerator and the denominator**:
   - For the numerator:  
     \[
     (5 + 7i)(1 - i) = 5(1) - 5(i) + 7i(1) - 7i^2 = 5 - 5i + 7i - 7(-1)
     \]
     Simplifying this:
     \[
     5 + 7 + 2i = 12 + 2i
     \]

   - For the denominator:
     \[
     (1 + i)(1 - i) = 1^2 - i^2 = 1 - (-1) = 1 + 1 = 2
     \]

3. **Write the final expression**:
   \[
   \frac{12 + 2i}{2} = \frac{12}{2} + \frac{2i}{2} = 6 + i
   \]

So, the simplified form of the expression \(\frac{5 + 7i}{1 + i}\) is \(\boxed{6 + i}\).

Would you like any further details on this, or do you have any other questions?

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Here are 5 related questions for further exploration:

1. What is the conjugate of a complex number and how is it used in division?
2. How can you verify the correctness of the result obtained by multiplying by the conjugate?
3. What is the geometric interpretation of multiplying a complex number by its conjugate?
4. How would you divide two complex numbers where both have imaginary parts?
5. Can you generalize this method to divide any two complex numbers?

**Tip:** When dividing complex numbers, always multiply by the conjugate of the denominator to eliminate the imaginary part from the denominator.

Learn how to simplify the division of complex numbers (5 + 7i) / (1 + i) step-by-step. Understand the concept of conjugates and how they are used in division of complex numbers. Suitable for advanced high school and college students.

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