**Simplify the following Expression**

6+3i/1+5i=

To simplify the expression \(\frac{6 + 3i}{1 + 5i}\), we'll use the method of multiplying both the numerator and denominator by the complex conjugate of the denominator.

### Steps:
1. The complex conjugate of \(1 + 5i\) is \(1 - 5i\).
2. Multiply both the numerator and the denominator by \(1 - 5i\):

\[
\frac{6 + 3i}{1 + 5i} \times \frac{1 - 5i}{1 - 5i}
\]

This gives:

\[
\frac{(6 + 3i)(1 - 5i)}{(1 + 5i)(1 - 5i)}
\]

3. Now simplify both the numerator and the denominator:

#### Numerator:
\[
(6 + 3i)(1 - 5i) = 6(1) + 6(-5i) + 3i(1) + 3i(-5i)
\]
\[
= 6 - 30i + 3i - 15i^2
\]
Since \(i^2 = -1\), replace \(i^2\) with \(-1\):
\[
= 6 - 30i + 3i + 15 = 21 - 27i
\]

#### Denominator:
\[
(1 + 5i)(1 - 5i) = 1^2 - (5i)^2 = 1 - 25i^2
\]
Since \(i^2 = -1\):
\[
= 1 + 25 = 26
\]

4. Now, the simplified form of the expression is:

\[
\frac{21 - 27i}{26}
\]

We can break this down further as:

\[
\frac{21}{26} - \frac{27}{26}i
\]

So, the simplified expression is:

\[
\boxed{\frac{21}{26} - \frac{27}{26}i}
\]

Would you like any additional details or further clarification on this?

Here are 5 related questions:
1. How do you find the complex conjugate of a complex number?
2. What is the importance of multiplying by the complex conjugate in these cases?
3. How does \(i^2 = -1\) affect the simplification of complex expressions?
4. Can you simplify \(\frac{4 + 7i}{2 + i}\) using a similar method?
5. What happens if you multiply two purely imaginary numbers?

**Tip:** When simplifying complex fractions, always multiply by the conjugate of the denominator to eliminate the imaginary part.

Simplify the following Expression 6+3i/1+5i=

Simplify the complex expression (6+3i)/(1+5i) by multiplying by the conjugate of the denominator. This problem covers key concepts of complex numbers, algebra, and conjugate multiplication, with detailed step-by-step guidance. Suitable for students in Grades 10-12.

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