Question

use the complex conjugate to divide these complex numbers.\\(\frac{5 + 4i}{6 - 2i}\\)\\(\frac{?}{?} + \frac{?}{?}i\\)remember to simplify.

Explanation:

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

The complex conjugate of \(6 - 2i\) is \(6 + 2i\). So we multiply the fraction by \(\frac{6 + 2i}{6 + 2i}\):
\[
\frac{(5 + 4i)(6 + 2i)}{(6 - 2i)(6 + 2i)}
\]

Step2: Expand the numerator and the denominator.

Expand the numerator using the distributive property (FOIL method):
\[

$$\begin{align*} (5 + 4i)(6 + 2i)&=5\times6 + 5\times2i + 4i\times6 + 4i\times2i\\ &= 30 + 10i + 24i + 8i^2 \end{align*}$$

\]
Since \(i^2=- 1\), we have:
\[
30 + 34i+8\times(-1)=30 + 34i - 8 = 22 + 34i
\]
Expand the denominator using the formula \((a - b)(a + b)=a^2 - b^2\):
\[
(6 - 2i)(6 + 2i)=6^2-(2i)^2=36 - 4i^2
\]
Again, since \(i^2 = - 1\), we get:
\[
36-4\times(-1)=36 + 4=40
\]

Step3: Simplify the fraction.

Now our fraction is \(\frac{22 + 34i}{40}\). We can split this into the real and imaginary parts:
\[
\frac{22}{40}+\frac{34}{40}i
\]
Simplify the fractions:
\[
\frac{11}{20}+\frac{17}{20}i
\]

Answer:

The real part is \(\frac{11}{20}\) and the imaginary part is \(\frac{17}{20}\), so the result is \(\frac{11}{20}+\frac{17}{20}i\)