Question

1. \frac{-8 - 2i}{4 + 6i}

Explanation:

Step1: Multiply numerator and denominator by conjugate of denominator.

The conjugate of \(4 + 6i\) is \(4 - 6i\). So we multiply the fraction \(\frac{-8 - 2i}{4 + 6i}\) by \(\frac{4 - 6i}{4 - 6i}\).

$$ \frac{(-8 - 2i)(4 - 6i)}{(4 + 6i)(4 - 6i)} $$

Step2: Expand numerator and denominator.

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

$$
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$$

Now, expand the denominator using the difference of squares formula \((a + b)(a - b)=a^2 - b^2\):

$$
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$$

Step3: Simplify the fraction.

Now we have \(\frac{-44 + 40i}{52}\). We can factor out a 4 from the numerator and denominator:

$$ \frac{4(-11 + 10i)}{4\times13}=\frac{-11 + 10i}{13}=-\frac{11}{13}+\frac{10}{13}i $$

Answer:

\(-\frac{11}{13}+\frac{10}{13}i\)