Question
Answer
$$\dfrac{1}{\dfrac{12-3i}{33000+22500-81750i}}=\dfrac{4339250}{362i+405}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{1}{\frac{12-3i}{33000+22500-81750i}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{1}{\frac{12-3i}{-81750i+55500}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{1}{\frac{405+362i}{4339250}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{4339250}{362i+405}\end{aligned} $$ | |
| ① | Simplify denominator $$ \color{blue}{33000} + \color{blue}{22500} -81750i = -81750i+ \color{blue}{55500} $$ |
| ② | Divide $ \, 12-3i \, $ by $ \, 55500-81750i \, $ to get $\,\, \dfrac{405+362i}{4339250} $. ( view steps ) |
| ③ | Divide $1$ by $ \dfrac{405+362i}{4339250} $ to get $ \dfrac{4339250}{362i+405} $. Step 1: To divide rational expressions, multiply the first fraction by the reciprocal of the second fraction. Step 2: Write $ 1 $ as a fraction by putting $ \color{red}{1} $ in the denominator. Step 3: Multiply numerators and denominators. Step 4: Simplify numerator and denominator. $$ \begin{aligned} \frac{1}{ \frac{\color{blue}{405+362i}}{\color{blue}{4339250}} } & \xlongequal{\text{Step 1}} 1 \cdot \frac{\color{blue}{4339250}}{\color{blue}{405+362i}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{1}{\color{red}{1}} \cdot \frac{4339250}{405+362i} \xlongequal{\text{Step 3}} \frac{ 1 \cdot 4339250 }{ 1 \cdot \left( 405+362i \right) } = \\[1ex] & \xlongequal{\text{Step 4}} \frac{ 4339250 }{ 405+362i } = \frac{4339250}{362i+405} \end{aligned} $$ |
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