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To understand complex conjugates, start with a complex number, which often looks like this: \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. The complex conjugate is formed by changing the sign of the imaginary part, resulting in \(a - bi\). This simple change plays a crucial role in simplifying complex number division.
Multiplying a complex number by its conjugate always yields a real number. This is due to a special formula, called the difference of squares:

• For any complex number \((x + yi)\), its product with its complex conjugate \((x - yi)\) is \( x^2 + y^2 \).
• This operation removes the imaginary unit \(i\) since \(i^2 = -1\), resulting in a neat and real output.

In our exercise, the complex conjugate of the denominator \(3 + i\) is \(3 - i\). By employing this technique, we transformed the division of complex numbers into an easier operation.

Dividing complex numbers might seem tricky at first. However, by using the complex conjugate of the denominator, we can simplify to achieve a straightforward result.
Here's how it works:

• The numerator and denominator are each multiplied by the conjugate of the denominator.
• For the expression \(\frac{2i}{3+i}\), both parts get multiplied by \(3-i\).

This maneuver doesn't change the value of the original expression but cleverly rearranges it to where our problematic denominator has transformed into a real number. With this application, reducing the complex fraction \(\frac{2i(3-i)}{(3+i)(3-i)}\) becomes straightforward, as we'll see in the simplification process.

Once the division is reformatted, simplification becomes an accessible task.
We proceed by simplifying separately the newly structured equation's numerator and denominator.For the numerator, \(2i(3-i)\), apply the distributive property:- \(2i \times 3 = 6i\)- \(2i \times (-i) = -2i^2\)Remember, \(i^2\) equates to \(-1\), transforming \(-2i^2\) to \(2\). So, the numerator, \(6i - 2i^2\), simplifies to \(6i + 2\).
For the denominator, \((3+i)(3-i)\), use the difference of squares:- \(3^2 - i^2 = 9 - (-1) = 10\)This simplified quotient \(\frac{6i+2}{10}\) is then separated into individual components:

• Real part: \(\frac{2}{10} = \frac{1}{5}\)
• Imaginary part: \(\frac{6i}{10} = \frac{3i}{5}\)

Finally, the fraction becomes \(\frac{1}{5} + \frac{3}{5}i\), offering a clearer and simpler perspective on complex number operations.