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 $\color{blue}{3}$.

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

Divide $i$ by $ \dfrac{-3i+2}{3} $ to get $ \dfrac{ 3i }{ -3i+2 } $.

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

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

Step 3: Multiply numerators and denominators.

Step 4: Simplify numerator and denominator.

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

Divide $ \, 3i \, $ by $ \, 2-3i \, $ to get $\,\, \dfrac{-9+6i}{13} $.
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