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Question

Answer

$$\dfrac{1}{-100i}+\dfrac{1}{9.9+99.01i}=\dfrac{50-i}{54900}$$

Explanation

Tap the blue circles to see an explanation.

$$ \begin{aligned}\frac{1}{-100i}+\frac{1}{9.9+99.01i}& \xlongequal{ }\frac{1}{-100i}+\frac{1}{9.9+99i} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{-i+9}{-9900i^2-900i} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{-i+9}{9900-900i} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{50-i}{54900}\end{aligned} $$
①

Add $ \dfrac{1}{-100i} $ and $ \dfrac{1}{9+99i} $ to get $ \dfrac{ \color{purple}{ -i+9 } }{ -9900i^2-900i }$.

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}{ 99i+9 }$ and the second by $\color{blue}{ -100i }$.

$$ \begin{aligned} \frac{1}{-100i} + \frac{1}{9+99i} & = \frac{ 1 \cdot \color{blue}{ \left( 99i+9 \right) }}{ \left( -100i \right) \cdot \color{blue}{ \left( 99i+9 \right) }} + \frac{ 1 \cdot \color{blue}{ \left( -100i \right) }}{ \left( 9+99i \right) \cdot \color{blue}{ \left( -100i \right) }} = \\[1ex] &=\frac{ \color{purple}{ 99i+9 } }{ -9900i^2-900i } + \frac{ \color{purple}{ -100i } }{ -9900i^2-900i } = \\[1ex] &=\frac{ \color{purple}{ -i+9 } }{ -9900i^2-900i } \end{aligned} $$
②
$$ -9900i^2 = -9900 \cdot (-1) = 9900 $$
③
Divide $ \, 9-i \, $ by $ \, 9900-900i \, $ to get $\,\, \dfrac{50-i}{54900} $.
( view steps )
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