Question

express as a complex number in simplest a+bi form: \\(\frac{-9 - 3i}{-3 + 2i}\\)

Explanation:

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

The conjugate of \(-3 + 2i\) is \(-3 - 2i\). So we have:

$$ \frac{(-9 - 3i)(-3 - 2i)}{(-3 + 2i)(-3 - 2i)} $$

Step2: Expand the numerator

Using the distributive property (FOIL method):

$$
LATEXBLOCK0
$$

Step3: Expand the denominator

Using the difference of squares formula \((a + b)(a - b)=a^2 - b^2\):

$$
LATEXBLOCK1
$$

Step4: Simplify the fraction

Now we have \(\frac{21 + 27i}{13}\), which can be written as:

$$ \frac{21}{13}+\frac{27}{13}i $$

Answer:

\(\frac{21}{13}+\frac{27}{13}i\)