Question
Answer
$$(51.72-20.69)\cdot10^4i\dfrac{x}{51.72-20.69i+10^4ix}=\dfrac{310000ix}{10000ix-20i+51}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(51.72-20.69)\cdot10^4i\frac{x}{51.72-20.69i+10^4ix}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}31\cdot10^4i\frac{x}{51.72-20.69i+10^4ix} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}31\cdot10000i\frac{x}{51.72-20.69i+10^4ix} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}310000i\frac{x}{51.72-20.69i+10^4ix} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}310000i\frac{x}{51.72-20i+10000ix} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{310000ix}{10000ix-20i+51}\end{aligned} $$ | |
| ① | Combine like terms: $$ \color{blue}{51} \color{blue}{-20} = \color{blue}{31} $$ |
| ② | i-i=0i |
| ③ | $$ 31 \cdot 10000 i = 310000 i $$ |
| ④ | 20i-20i=0i |
| ⑤ | Multiply $310000i$ by $ \dfrac{x}{51-20i+10000ix} $ to get $ \dfrac{310000ix}{10000ix-20i+51} $. Step 1: Write $ 310000i $ as a fraction by putting $ \color{red}{1} $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} 310000i \cdot \frac{x}{51-20i+10000ix} & \xlongequal{\text{Step 1}} \frac{310000i}{\color{red}{1}} \cdot \frac{x}{51-20i+10000ix} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ 310000i \cdot x }{ 1 \cdot \left( 51-20i+10000ix \right) } \xlongequal{\text{Step 3}} \frac{ 310000ix }{ 51-20i+10000ix } = \\[1ex] &= \frac{310000ix}{10000ix-20i+51} \end{aligned} $$ |
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