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 \(b = 2i\):
\[
(6 - 2i)(6 + 2i)=6^2-(2i)^2=36 - 4i^2
\]
Substitute \(i^2 = - 1\):
\[
36 - 4\times(-1)=36 + 4 = 40
\]

Step3: Simplify the fraction.

Now our fraction becomes \(\frac{22 + 34i}{40}\). We can split this into two fractions:
\[
\frac{22}{40}+\frac{34}{40}i
\]
Simplify the fractions by dividing numerator and denominator by their greatest common divisors. For \(\frac{22}{40}\), the GCD of 22 and 40 is 2, so \(\frac{22\div2}{40\div2}=\frac{11}{20}\). For \(\frac{34}{40}\), the GCD of 34 and 40 is 2, so \(\frac{34\div2}{40\div2}=\frac{17}{20}\).
So the simplified form is \(\frac{11}{20}+\frac{17}{20}i\).

Answer:

The real part is \(\frac{11}{20}\) and the imaginary part is \(\frac{17}{20}\), so the answer is \(\frac{11}{20}+\frac{17}{20}i\) (with the real part numerator 11, real part denominator 20, imaginary part numerator 17, imaginary part denominator 20).