Question

use the complex conjugate to divide these complex numbers.\\(\frac{9 + 2i}{6 - 3i}\\)\\(\frac{?}{\box} + \frac{\box}{\box}i\\)\
remember to simplify.

Explanation:

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

The complex conjugate of \(6 - 3i\) is \(6 + 3i\). So we multiply the fraction \(\frac{9 + 2i}{6 - 3i}\) by \(\frac{6 + 3i}{6 + 3i}\):
\[
\frac{(9 + 2i)(6 + 3i)}{(6 - 3i)(6 + 3i)}
\]

Step2: Expand the numerator and the denominator.

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

$$\begin{align*} (9 + 2i)(6 + 3i)&=9\times6 + 9\times3i + 2i\times6 + 2i\times3i\\ &= 54 + 27i + 12i + 6i^2 \end{align*}$$

\]
Since \(i^2=- 1\), we substitute that in:
\[
54 + 27i + 12i+6\times(-1)=54 + 39i - 6 = 48 + 39i
\]
Now expand the denominator using the difference of squares formula \((a - b)(a + b)=a^2 - b^2\), where \(a = 6\) and \(b = 3i\):
\[
(6 - 3i)(6 + 3i)=6^2-(3i)^2=36 - 9i^2
\]
Substitute \(i^2 = - 1\):
\[
36-9\times(-1)=36 + 9=45
\]
So now our fraction is \(\frac{48 + 39i}{45}\)

Step3: Simplify the fraction.

We can factor out a 3 from the numerator: \(\frac{3(16 + 13i)}{45}=\frac{16 + 13i}{15}\)
Then we split the fraction into the real and imaginary parts: \(\frac{16}{15}+\frac{13}{15}i\)

Answer:

The real part is \(\frac{16}{15}\) and the imaginary part is \(\frac{13}{15}\), so the answer is \(\frac{16}{15}+\frac{13}{15}i\) (the real part numerator is 16, real part denominator is 15, imaginary part numerator is 13, imaginary part denominator is 15).