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Question

Answer

$$\dfrac{2}{\dfrac{1}{5}-\dfrac{1}{2}i+\dfrac{1}{5}}=\dfrac{20}{-5i+4}$$

Explanation

Tap the blue circles to see an explanation.

$$ \begin{aligned}\frac{2}{\frac{1}{5}-\frac{1}{2}i+\frac{1}{5}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{2}{\frac{1}{5}-\frac{i}{2}+\frac{1}{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{2}{\frac{-5i+2}{10}+\frac{1}{5}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{2}{\frac{-5i+4}{10}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{20}{-5i+4}\end{aligned} $$
①

Multiply $ \dfrac{1}{2} $ by $ i $ to get $ \dfrac{ i }{ 2 } $.

Step 1: Write $ i $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{1}{2} \cdot i & \xlongequal{\text{Step 1}} \frac{1}{2} \cdot \frac{i}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 1 \cdot i }{ 2 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ i }{ 2 } \end{aligned} $$
②

Subtract $ \dfrac{i}{2} $ from $ \dfrac{1}{5} $ to get $ \dfrac{ \color{purple}{ -5i+2 } }{ 10 }$.

To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $ \color{blue}{ 2 }$ and the second by $\color{blue}{ 5 }$.

$$ \begin{aligned} \frac{1}{5} - \frac{i}{2} & = \frac{ 1 \cdot \color{blue}{ 2 }}{ 5 \cdot \color{blue}{ 2 }} - \frac{ i \cdot \color{blue}{ 5 }}{ 2 \cdot \color{blue}{ 5 }} = \\[1ex] &=\frac{ \color{purple}{ 2 } }{ 10 } - \frac{ \color{purple}{ 5i } }{ 10 }=\frac{ \color{purple}{ -5i+2 } }{ 10 } \end{aligned} $$
③

Add $ \dfrac{-5i+2}{10} $ and $ \dfrac{1}{5} $ to get $ \dfrac{ \color{purple}{ -5i+4 } }{ 10 }$.

To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{2}$.

$$ \begin{aligned} \frac{-5i+2}{10} + \frac{1}{5} & = \frac{ -5i+2 }{ 10 } + \frac{ 1 \cdot \color{blue}{ 2 }}{ 5 \cdot \color{blue}{ 2 }} = \\[1ex] &=\frac{ \color{purple}{ -5i+2 } }{ 10 } + \frac{ \color{purple}{ 2 } }{ 10 }=\frac{ \color{purple}{ -5i+4 } }{ 10 } \end{aligned} $$
④

Divide $2$ by $ \dfrac{-5i+4}{10} $ to get $ \dfrac{ 20 }{ -5i+4 } $.

Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction.

Step 2: Write $ 2 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 3: Multiply numerators and denominators.

Step 4: Simplify numerator and denominator.

$$ \begin{aligned} \frac{2}{ \frac{\color{blue}{-5i+4}}{\color{blue}{10}} } & \xlongequal{\text{Step 1}} 2 \cdot \frac{\color{blue}{10}}{\color{blue}{-5i+4}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{2}{\color{red}{1}} \cdot \frac{10}{-5i+4} \xlongequal{\text{Step 3}} \frac{ 2 \cdot 10 }{ 1 \cdot \left( -5i+4 \right) } = \\[1ex] & \xlongequal{\text{Step 4}} \frac{ 20 }{ -5i+4 } \end{aligned} $$
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