Answer
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{1}{\frac{1}{\frac{3}{5}-\frac{24}{5}i}+\frac{1}{4}}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{1}{\frac{1}{\frac{3}{5}-\frac{24i}{5}}+\frac{1}{4}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{1}{\frac{1}{\frac{-24i+3}{5}}+\frac{1}{4}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{1}{\frac{5}{-24i+3}+\frac{1}{4}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{1}{\frac{-24i+23}{-96i+12}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}\frac{1}{\frac{43+32i}{156}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}\frac{156}{32i+43}\end{aligned} $$ | |
| ① | Multiply $ \dfrac{24}{5} $ by $ i $ to get $ \dfrac{ 24i }{ 5 } $. Step 1: Write $ i $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} \frac{24}{5} \cdot i & \xlongequal{\text{Step 1}} \frac{24}{5} \cdot \frac{i}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 24 \cdot i }{ 5 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 24i }{ 5 } \end{aligned} $$ |
| ② | Subtract $ \dfrac{24i}{5} $ from $ \dfrac{3}{5} $ to get $ \dfrac{-24i+3}{5} $. To subtract expressions with the same denominators, we subtract the numerators and write the result over the common denominator. $$ \begin{aligned} \frac{3}{5} - \frac{24i}{5} & = \frac{3}{\color{blue}{5}} - \frac{24i}{\color{blue}{5}} =\frac{ 3 - 24i }{ \color{blue}{ 5 }} = \\[1ex] &= \frac{-24i+3}{5} \end{aligned} $$ |
| ③ | Divide $1$ by $ \dfrac{-24i+3}{5} $ to get $ \dfrac{ 5 }{ -24i+3 } $. 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. $$ \begin{aligned} \frac{1}{ \frac{\color{blue}{-24i+3}}{\color{blue}{5}} } & \xlongequal{\text{Step 1}} 1 \cdot \frac{\color{blue}{5}}{\color{blue}{-24i+3}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{1}{\color{red}{1}} \cdot \frac{5}{-24i+3} \xlongequal{\text{Step 3}} \frac{ 1 \cdot 5 }{ 1 \cdot \left( -24i+3 \right) } = \\[1ex] & \xlongequal{\text{Step 4}} \frac{ 5 }{ -24i+3 } \end{aligned} $$ |
| ④ | Add $ \dfrac{5}{-24i+3} $ and $ \dfrac{1}{4} $ to get $ \dfrac{ \color{purple}{ -24i+23 } }{ -96i+12 }$. To add raitonal expressions, both fractions must have the same denominator. |
| ⑤ | Divide $ \, 23-24i \, $ by $ \, 12-96i \, $ to get $\,\, \dfrac{43+32i}{156} $. ( view steps ) |
| ⑥ | Divide $1$ by $ \dfrac{43+32i}{156} $ to get $ \dfrac{156}{32i+43} $. 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. $$ \begin{aligned} \frac{1}{ \frac{\color{blue}{43+32i}}{\color{blue}{156}} } & \xlongequal{\text{Step 1}} 1 \cdot \frac{\color{blue}{156}}{\color{blue}{43+32i}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{1}{\color{red}{1}} \cdot \frac{156}{43+32i} \xlongequal{\text{Step 3}} \frac{ 1 \cdot 156 }{ 1 \cdot \left( 43+32i \right) } = \\[1ex] & \xlongequal{\text{Step 4}} \frac{ 156 }{ 43+32i } = \frac{156}{32i+43} \end{aligned} $$ |