Question

use the complex conjugate to divide these complex numbers.\\(\frac{5 + 3i}{8 - 2i}\\)\\(\frac{?}{\square}+\frac{\square}{\square}i\\)remember to simplify.

Explanation:

Step1: Multiply numerator and denominator by the complex conjugate of the denominator.

The complex conjugate of \(8 - 2i\) is \(8 + 2i\). So we multiply the fraction \(\frac{5 + 3i}{8 - 2i}\) by \(\frac{8 + 2i}{8 + 2i}\):
\[
\frac{(5 + 3i)(8 + 2i)}{(8 - 2i)(8 + 2i)}
\]

Step2: Expand the numerator and the denominator.

First, expand the numerator using the distributive property (FOIL method):
\[

$$\begin{align*} (5 + 3i)(8 + 2i)&=5\times8 + 5\times2i + 3i\times8 + 3i\times2i\\ &=40 + 10i + 24i + 6i^2 \end{align*}$$

\]
Since \(i^2=-1\), we substitute that in:
\[
40 + 10i + 24i + 6\times(-1)=40 + 34i - 6 = 34 + 34i
\]
Now, expand the denominator using the difference of squares formula \((a - b)(a + b)=a^2 - b^2\), where \(a = 8\) and \(b = 2i\):
\[

$$\begin{align*} (8 - 2i)(8 + 2i)&=8^2 - (2i)^2\\ &=64 - 4i^2 \end{align*}$$

\]
Substitute \(i^2 = - 1\):
\[
64 - 4\times(-1)=64 + 4 = 68
\]

Step3: Simplify the fraction.

Now our fraction is \(\frac{34 + 34i}{68}\). We can split this into two fractions:
\[
\frac{34}{68}+\frac{34i}{68}
\]
Simplify each fraction:
\[
\frac{1}{2}+\frac{1}{2}i
\]

Answer:

\(\frac{1}{2} + \frac{1}{2}i\) (So the real part is \(\frac{1}{2}\) and the imaginary part is \(\frac{1}{2}\))