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Answer

$$\frac{30.07-10.94i}{(6-8i)\cdot(4+2i)}+\frac{20}{-10+24i}=\frac{683-1031i}{1690}$$

Explanation

Tap the blue circles to see an explanation.

$$ \begin{aligned}\frac{30.07-10.94i}{(6-8i)\cdot(4+2i)}+\frac{20}{-10+24i}& \xlongequal{ }\frac{30.07-10i}{(6-8i)\cdot(4+2i)}+\frac{20}{-10+24i} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{30.07-10i}{24+12i-32i-16i^2}+\frac{20}{-10+24i} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{30.07-10i}{-16i^2-20i+24}+\frac{20}{-10+24i} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{30.07-10i}{16-20i+24}+\frac{20}{-10+24i} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{30.07-10i}{-20i+40}+\frac{20}{-10+24i} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{7+i}{10}+\frac{20}{-10+24i} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}\frac{24i^2+158i+130}{240i-100} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}\frac{-24+158i+130}{240i-100} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}\frac{158i+106}{240i-100} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle9}{\textcircled {9}} } }}}\frac{683-1031i}{1690}\end{aligned} $$
①

Multiply each term of $ \left( \color{blue}{6-8i}\right) $ by each term in $ \left( 4+2i\right) $.

$$ \left( \color{blue}{6-8i}\right) \cdot \left( 4+2i\right) = 24+12i-32i-16i^2 $$
②

Combine like terms:

$$ 24+ \color{blue}{12i} \color{blue}{-32i} -16i^2 = -16i^2 \color{blue}{-20i} +24 $$
③
$$ -16i^2 = -16 \cdot (-1) = 16 $$
④

Combine like terms:

$$ \color{blue}{16} -20i+ \color{blue}{24} = -20i+ \color{blue}{40} $$
⑤
Divide $ \, 30-10i \, $ by $ \, 40-20i \, $ to get $\,\, \dfrac{7+i}{10} $.
( view steps )
⑥

Add $ \dfrac{7+i}{10} $ and $ \dfrac{20}{-10+24i} $ to get $ \dfrac{ \color{purple}{ 24i^2+158i+130 } }{ 240i-100 }$.

To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $ \color{blue}{ 24i-10 }$ and the second by $\color{blue}{ 10 }$.

$$ \begin{aligned} \frac{7+i}{10} + \frac{20}{-10+24i} & = \frac{ \left( 7+i \right) \cdot \color{blue}{ \left( 24i-10 \right) }}{ 10 \cdot \color{blue}{ \left( 24i-10 \right) }} + \frac{ 20 \cdot \color{blue}{ 10 }}{ \left( -10+24i \right) \cdot \color{blue}{ 10 }} = \\[1ex] &=\frac{ \color{purple}{ 168i-70+24i^2-10i } }{ 240i-100 } + \frac{ \color{purple}{ 200 } }{ 240i-100 } = \\[1ex] &=\frac{ \color{purple}{ 24i^2+158i+130 } }{ 240i-100 } \end{aligned} $$
⑦
$$ 24i^2 = 24 \cdot (-1) = -24 $$
⑧

Simplify numerator

$$ \color{blue}{-24} +158i+ \color{blue}{130} = 158i+ \color{blue}{106} $$
⑨
Divide $ \, 106+158i \, $ by $ \, -100+240i \, $ to get $\,\, \dfrac{683-1031i}{1690} $.
( view steps )
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