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To solve the expression \(\frac{3 + 4i}{1 + i}\), we need to divide these complex numbers and write the result in standard form \(a + bi\), where \(a\) and \(b\) are real numbers.

### Step-by-step solution:

1. **Multiply by the conjugate**: To simplify the division, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(1 + i\) is \(1 - i\).
   \[
   \frac{3 + 4i}{1 + i} \cdot \frac{1 - i}{1 - i} = \frac{(3 + 4i)(1 - i)}{(1 + i)(1 - i)}
   \]

2. **Simplify the denominator**: Use the difference of squares formula:
   \[
   (1 + i)(1 - i) = 1^2 - i^2 = 1 - (-1) = 2
   \]
   So, the denominator simplifies to \(2\).

3. **Multiply the numerator**: Now, distribute \( (3 + 4i)(1 - i) \):
   \[
   3(1 - i) + 4i(1 - i) = 3 - 3i + 4i - 4i^2
   \]
   Simplifying:
   \[
   3 - 3i + 4i - 4(-1) = 3 - 3i + 4i + 4 = 7 + i
   \]

4. **Final division**: Now divide the result by \(2\):
   \[
   \frac{7 + i}{2} = \frac{7}{2} + \frac{1}{2}i
   \]

Thus, the final answer in standard form is:
\[
\frac{7}{2} + \frac{1}{2}i \quad \text{or} \quad 3.5 + 0.5i
\]

Would you like more details on any step or have any questions? Here are some related questions to explore further:

1. What is the significance of the conjugate when dividing complex numbers?
2. How do you multiply two complex numbers in general?
3. What are some real-world applications of complex numbers?
4. Can you divide complex numbers without using conjugates?
5. How do we interpret complex numbers geometrically on the complex plane?

**Tip**: Always remember to rationalize the denominator by multiplying with the conjugate when dividing complex numbers.

Divide (3 + 4i) by (1 + i) and write the answer in standard form.

Learn how to divide complex numbers like (3 + 4i) ÷ (1 + i) using the conjugate of the denominator. Follow a detailed step-by-step solution and understand key concepts in complex numbers for students in Grades 10-12.

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