Question

express as a complex number in simplest a+bi form: \\(\frac{13 + 9i}{-2 - i}\\)

Explanation:

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

The conjugate of \(-2 - i\) is \(-2 + i\). So we have:

$$ \frac{(13 + 9i)(-2 + i)}{(-2 - i)(-2 + i)} $$

Step2: Expand the numerator and the denominator.

First, expand the numerator:

$$
LATEXBLOCK0
$$

Then, expand the denominator using the difference of squares formula \((a - b)(a + b)=a^{2}-b^{2}\), where \(a=-2\) and \(b = i\):

$$
LATEXBLOCK1
$$

Step3: Divide the expanded numerator by the expanded denominator.

$$ \frac{-35 - 5i}{5}=\frac{-35}{5}+\frac{-5i}{5}=-7 - i $$

Answer:

\(-7 - i\)