Answer
$$ai\frac{b}{a+ib}=\frac{abi}{bi+a}$$
Explanation
| $$ \begin{aligned}ai\frac{b}{a+ib}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{abi}{bi+a}\end{aligned} $$ | |
| ① | Multiply $ai$ by $ \dfrac{b}{a+bi} $ to get $ \dfrac{abi}{bi+a} $. Step 1: Write $ ai $ as a fraction by putting $ \color{red}{1} $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} ai \cdot \frac{b}{a+bi} & \xlongequal{\text{Step 1}} \frac{ai}{\color{red}{1}} \cdot \frac{b}{a+bi} \xlongequal{\text{Step 2}} \frac{ ai \cdot b }{ 1 \cdot \left( a+bi \right) } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ abi }{ a+bi } = \frac{abi}{bi+a} \end{aligned} $$ |
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