Question

use the complex conjugate to divide these complex numbers.\\(\frac{5 + 4i}{6 - 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 \(6 - 2i\) is \(6 + 2i\). So we multiply the fraction \(\frac{5 + 4i}{6 - 2i}\) by \(\frac{6 + 2i}{6 + 2i}\):
\[
\frac{(5 + 4i)(6 + 2i)}{(6 - 2i)(6 + 2i)}
\]

Step2: Expand the numerator and the denominator.

First, 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 substitute that in:
\[
30 + 10i + 24i + 8\times(-1)=30 + 34i - 8 = 22 + 34i
\]
Now, expand the denominator using the difference of squares formula \((a - b)(a + b)=a^2 - b^2\), where \(a = 6\) and \(b = 2i\):
\[

$$\begin{align*} (6 - 2i)(6 + 2i)&=6^2-(2i)^2\\ &=36 - 4i^2 \end{align*}$$

\]
Substitute \(i^2 = - 1\):
\[
36 - 4\times(-1)=36 + 4 = 40
\]

Step3: Write the fraction as a complex number in standard form.

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:

\(\frac{11}{20}+\frac{17}{20}i\) (So the real part numerator is \(11\), real part denominator is \(20\), imaginary part numerator is \(17\), imaginary part denominator is \(20\))