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Let's simplify the given expressions:

### a. \(\frac{4 + i}{7 + i}\)

To simplify this complex fraction, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(7 + i\) is \(7 - i\).

\[
\frac{4 + i}{7 + i} \times \frac{7 - i}{7 - i} = \frac{(4 + i)(7 - i)}{(7 + i)(7 - i)}
\]

First, simplify the denominator:

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

Now, expand the numerator:

\[
(4 + i)(7 - i) = 4 \times 7 - 4 \times i + i \times 7 - i \times i = 28 - 4i + 7i - i^2
\]

Simplify the expression:

\[
= 28 + 3i + 1 = 29 + 3i
\]

So, the simplified form is:

\[
\frac{4 + i}{7 + i} = \frac{29 + 3i}{50} = \frac{29}{50} + \frac{3i}{50}
\]

### b. \((12 + 6i) + (9 + 5i)\)

To simplify this expression, just add the real parts and the imaginary parts separately:

\[
(12 + 6i) + (9 + 5i) = (12 + 9) + (6i + 5i) = 21 + 11i
\]

So the simplified expression is:

\[
21 + 11i
\]

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Would you like more details or have any questions? Here are some related problems for practice:

1. Simplify \(\frac{3 + 2i}{4 - 3i}\).
2. Add \((8 + 2i) + (5 - 3i)\).
3. Simplify \(\frac{5 - 4i}{2 + 3i}\).
4. Subtract \((7 + 9i) - (3 + 2i)\).
5. Multiply \((2 + 3i)(4 - i)\).

**Tip:** When dealing with complex numbers, always remember to simplify the denominator by multiplying by the conjugate to eliminate the imaginary part in the denominator.

Simplify the following expressions: a. (4 + i)/(7 + i) b. (12 + 6i) + (9 + 5i)

Learn how to simplify complex expressions such as (4 + i)/(7 + i) and (12 + 6i) + (9 + 5i). This guide covers key concepts in complex numbers, including the use of conjugates and addition of complex numbers. Suitable for students in Grades 9-12.

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