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Question

Answer

$$3+8i-4\cdot\dfrac{2-i}{3}-4i=\dfrac{16i+1}{3}$$

Explanation

Tap the blue circles to see an explanation.

$$ \begin{aligned}3+8i-4 \cdot \frac{2-i}{3}-4i& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}3+8i-\frac{-4i+8}{3}-4i \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{28i+1}{3}-4i \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{16i+1}{3}\end{aligned} $$
①

Multiply $4$ by $ \dfrac{2-i}{3} $ to get $ \dfrac{-4i+8}{3} $.

Step 1: Write $ 4 $ as a fraction by putting $ \color{red}{1} $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} 4 \cdot \frac{2-i}{3} & \xlongequal{\text{Step 1}} \frac{4}{\color{red}{1}} \cdot \frac{2-i}{3} \xlongequal{\text{Step 2}} \frac{ 4 \cdot \left( 2-i \right) }{ 1 \cdot 3 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 8-4i }{ 3 } = \frac{-4i+8}{3} \end{aligned} $$
②

Subtract $ \dfrac{-4i+8}{3} $ from $ 3+8i $ to get $ \dfrac{ \color{purple}{ 28i+1 } }{ 3 }$.

Step 1: Write $ 3+8i $ 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 first fraction by $\color{blue}{ 3 }$.

$$ \begin{aligned} 3+8i- \frac{-4i+8}{3} & \xlongequal{\text{Step 1}} \frac{3+8i}{\color{red}{1}} - \frac{-4i+8}{3} = \frac{ \left( 3+8i \right) \cdot \color{blue}{ 3 }}{ 1 \cdot \color{blue}{ 3 }} - \frac{ -4i+8 }{ 3 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 9+24i } }{ 3 } - \frac{ \color{purple}{ -4i+8 } }{ 3 }=\frac{ \color{purple}{ 9+24i - \left( -4i+8 \right) } }{ 3 } = \\[1ex] &=\frac{ \color{purple}{ 28i+1 } }{ 3 } \end{aligned} $$
③

Subtract $4i$ from $ \dfrac{28i+1}{3} $ to get $ \dfrac{ \color{purple}{ 16i+1 } }{ 3 }$.

Step 1: Write $ 4i $ 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 $\color{blue}{3}$.

$$ \begin{aligned} \frac{28i+1}{3} -4i & \xlongequal{\text{Step 1}} \frac{28i+1}{3} - \frac{4i}{\color{red}{1}} = \frac{ 28i+1 }{ 3 } - \frac{ 4i \cdot \color{blue}{ 3 }}{ 1 \cdot \color{blue}{ 3 }} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 28i+1 } }{ 3 } - \frac{ \color{purple}{ 12i } }{ 3 }=\frac{ \color{purple}{ 16i+1 } }{ 3 } \end{aligned} $$
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