Question

a complex number in simplest a+bi form: \\(\frac{-9 + 8i}{-3 + 9i}\\)

Explanation:

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

The conjugate of \(-3 + 9i\) is \(-3 - 9i\). So we have:

$$ \frac{(-9 + 8i)(-3 - 9i)}{(-3 + 9i)(-3 - 9i)} $$

Step2: Expand the numerator.

Using the distributive property (FOIL method):

$$
LATEXBLOCK0
$$

Step3: Expand the denominator.

Using the formula \((a + b)(a - b)=a^{2}-b^{2}\), where \(a=-3\) and \(b = 9i\):

$$ (-3)^{2}-(9i)^{2}=9 - 81i^{2}=9 - 81(-1)=9 + 81 = 90 $$

Step4: Simplify the fraction.

Divide the numerator by the denominator:

$$ \frac{99 + 57i}{90}=\frac{99}{90}+\frac{57}{90}i=\frac{11}{10}+\frac{19}{30}i $$

Answer:

\(\frac{11}{10}+\frac{19}{30}i\)