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Question

Answer

$$\dfrac{\dfrac{2}{3}-i}{2i+\dfrac{1}{2}}=\dfrac{-20-22i}{51}$$

Explanation

Tap the blue circles to see an explanation.

$$ \begin{aligned}\frac{\frac{2}{3}-i}{2i+\frac{1}{2}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{\frac{-3i+2}{3}}{\frac{4i+1}{2}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{-6i+4}{12i+3} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-20-22i}{51}\end{aligned} $$
①

Subtract $i$ from $ \dfrac{2}{3} $ to get $ \dfrac{ \color{purple}{ -3i+2 } }{ 3 }$.

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

Step 2: To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{3}$.

$$ \begin{aligned} \frac{2}{3} -i & \xlongequal{\text{Step 1}} \frac{2}{3} - \frac{i}{\color{red}{1}} = \frac{ 2 }{ 3 } - \frac{ i \cdot \color{blue}{ 3 }}{ 1 \cdot \color{blue}{ 3 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 2 } }{ 3 } - \frac{ \color{purple}{ 3i } }{ 3 }=\frac{ \color{purple}{ -3i+2 } }{ 3 } \end{aligned} $$
②

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

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

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

$$ \begin{aligned} 2i+ \frac{1}{2} & \xlongequal{\text{Step 1}} \frac{2i}{\color{red}{1}} + \frac{1}{2} = \frac{ 2i \cdot \color{blue}{ 2 }}{ 1 \cdot \color{blue}{ 2 }} + \frac{ 1 }{ 2 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 4i } }{ 2 } + \frac{ \color{purple}{ 1 } }{ 2 }=\frac{ \color{purple}{ 4i+1 } }{ 2 } \end{aligned} $$
③

Divide $ \dfrac{-3i+2}{3} $ by $ \dfrac{4i+1}{2} $ to get $ \dfrac{ -6i+4 }{ 12i+3 } $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{ \frac{-3i+2}{3} }{ \frac{\color{blue}{4i+1}}{\color{blue}{2}} } & \xlongequal{\text{Step 1}} \frac{-3i+2}{3} \cdot \frac{\color{blue}{2}}{\color{blue}{4i+1}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( -3i+2 \right) \cdot 2 }{ 3 \cdot \left( 4i+1 \right) } \xlongequal{\text{Step 3}} \frac{ -6i+4 }{ 12i+3 } \end{aligned} $$
④
Divide $ \, 4-6i \, $ by $ \, 3+12i \, $ to get $\,\, \dfrac{-20-22i}{51} $.
( view steps )
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