$$\dfrac{\dfrac{62500}{400i+300}}{250+500i+\dfrac{62500}{400i+300}}=\dfrac{62500}{250000i-62500}$$
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{\frac{62500}{400i+300}}{250+500i+\frac{62500}{400i+300}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{\frac{62500}{400i+300}}{\frac{200000i^2+250000i+137500}{400i+300}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{\frac{62500}{400i+300}}{\frac{-200000+250000i+137500}{400i+300}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{\frac{62500}{400i+300}}{\frac{250000i-62500}{400i+300}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{\frac{62500}{400i+300}}{325+400i} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{62500}{160000i^2+250000i+97500} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}\frac{62500}{-160000+250000i+97500} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}\frac{62500}{250000i-62500}\end{aligned} $$ | |
| ① | Add $250+500i$ and $ \dfrac{62500}{400i+300} $ to get $ \dfrac{ \color{purple}{ 200000i^2+250000i+137500 } }{ 400i+300 }$. Step 1: Write $ 250+500i $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: To add raitonal expressions, both fractions must have the same denominator. |
| ② | $$ 200000i^2 = 200000 \cdot (-1) = -200000 $$ |
| ③ | Combine like terms: $$ \color{blue}{-200000} +250000i+ \color{blue}{137500} = 250000i \color{blue}{-62500} $$ |
| ④ | Divide $ \, -62500+250000i \, $ by $ \, 300+400i \, $ to get $\,\, 325+400i $. ( view steps ) |
| ⑤ | Divide $ \dfrac{62500}{400i+300} $ by $ 325+400i $ to get $ \dfrac{62500}{160000i^2+250000i+97500} $. 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. $$ \begin{aligned} \frac{ \frac{62500}{400i+300} }{325+400i} & \xlongequal{\text{Step 1}} \frac{62500}{400i+300} \cdot \frac{\color{blue}{1}}{\color{blue}{325+400i}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ 62500 \cdot 1 }{ \left( 400i+300 \right) \cdot \left( 325+400i \right) } \xlongequal{\text{Step 3}} \frac{ 62500 }{ 130000i+160000i^2+97500+120000i } = \\[1ex] &= \frac{62500}{160000i^2+250000i+97500} \end{aligned} $$ |
| ⑥ | $$ 160000i^2 = 160000 \cdot (-1) = -160000 $$ |
| ⑦ | Simplify denominator $$ \color{blue}{-160000} +250000i+ \color{blue}{97500} = 250000i \color{blue}{-62500} $$ |