To simplify the expression \(\frac{6+3i}{2-8i}\), we'll multiply both the numerator and denominator by the complex conjugate of the denominator. Here's how:

1. The complex conjugate of \(2-8i\) is \(2+8i\).
2. Multiply both the numerator and denominator by \(2+8i\):

\[
\frac{6+3i}{2-8i} \times \frac{2+8i}{2+8i} = \frac{(6+3i)(2+8i)}{(2-8i)(2+8i)}
\]

Now let's simplify step by step:

### Step 1: Simplify the denominator
\[
(2-8i)(2+8i) = 2^2 - (8i)^2 = 4 - (-64) = 4 + 64 = 68
\]

### Step 2: Simplify the numerator
\[
(6+3i)(2+8i) = 6(2) + 6(8i) + 3i(2) + 3i(8i)
\]
\[
= 12 + 48i + 6i + 24i^2
\]
Since \(i^2 = -1\), replace \(24i^2\) with \(-24\):
\[
12 + 48i + 6i - 24 = (12 - 24) + (48i + 6i) = -12 + 54i
\]

### Step 3: Combine the results
\[
\frac{(6+3i)(2+8i)}{(2-8i)(2+8i)} = \frac{-12 + 54i}{68}
\]

Now divide both terms by 68:
\[
= \frac{-12}{68} + \frac{54i}{68} = -\frac{3}{17} + \frac{27i}{34}
\]

Thus, the simplified expression is:
\[
-\frac{3}{17} + \frac{27i}{34}
\]

Would you like more details or have any further questions?

Here are 5 related questions you might consider:
1. How do you find the complex conjugate of a complex number?
2. Why is multiplying by the complex conjugate used to simplify division of complex numbers?
3. What are some common applications of complex numbers in real-world scenarios?
4. Can you demonstrate simplifying other complex division problems?
5. How does the polar form of a complex number compare to its rectangular form?

**Tip:** When dividing complex numbers, always use the conjugate to eliminate the imaginary part in the denominator.

Explore how to simplify the complex division (6+3i)/(2-8i) step by step. Learn about key concepts such as complex conjugates, imaginary unit properties, and how to divide complex numbers. Suitable for high school students in Grades 10-12.

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