$$(5+1.2\cdot3.14i)\dfrac{1-i\cdot\dfrac{416.7}{3.14}}{6-i(\dfrac{416.7}{3.14}+1.2\cdot3.14)}=\dfrac{i+29225890}{17618409}$$
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(5+1.2\cdot3.14i)\frac{1-i\cdot\frac{416.7}{3.14}}{6-i(\frac{416.7}{3.14}+1.2\cdot3.14)}& \xlongequal{ }(5+3i)\frac{1-i\cdot\frac{416.7}{3.14}}{6-i(\frac{416.7}{3.14}+1.2\cdot3.14)} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}(5+3i)\frac{1-\frac{416i}{3}}{6-i\cdot\frac{4197}{10}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}(5+3i)\frac{\frac{-416i+3}{3}}{6-\frac{4197i}{10}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}(5+3i)\frac{\frac{-416i+3}{3}}{\frac{-4197i+60}{10}} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}(5+3i)\frac{-4160i+30}{-12591i+180} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}(5+3i)\frac{5820440-41230i}{17618409} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle9}{\textcircled {9}} } }}}\frac{-123690i^2+17255170i+29102200}{17618409} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle10}{\textcircled {10}} } }}}\frac{123690+i+29102200}{17618409} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle11}{\textcircled {11}} } }}}\frac{i+29225890}{17618409}\end{aligned} $$ | |
| ① | Multiply $i$ by $ \dfrac{416}{3} $ to get $ \dfrac{ 416i }{ 3 } $. 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. $$ \begin{aligned} i \cdot \frac{416}{3} & \xlongequal{\text{Step 1}} \frac{i}{\color{red}{1}} \cdot \frac{416}{3} \xlongequal{\text{Step 2}} \frac{ i \cdot 416 }{ 1 \cdot 3 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 416i }{ 3 } \end{aligned} $$ |
| ② | Combine like terms |
| ③ | Subtract $ \dfrac{416i}{3} $ from $ 1 $ to get $ \dfrac{ \color{purple}{ -416i+3 } }{ 3 }$. Step 1: Write $ 1 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: To subtract raitonal expressions, both fractions must have the same denominator. |
| ④ | Multiply $i$ by $ \dfrac{4197}{10} $ to get $ \dfrac{ 4197i }{ 10 } $. 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. $$ \begin{aligned} i \cdot \frac{4197}{10} & \xlongequal{\text{Step 1}} \frac{i}{\color{red}{1}} \cdot \frac{4197}{10} \xlongequal{\text{Step 2}} \frac{ i \cdot 4197 }{ 1 \cdot 10 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 4197i }{ 10 } \end{aligned} $$ |
| ⑤ | Subtract $ \dfrac{416i}{3} $ from $ 1 $ to get $ \dfrac{ \color{purple}{ -416i+3 } }{ 3 }$. Step 1: Write $ 1 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: To subtract raitonal expressions, both fractions must have the same denominator. |
| ⑥ | Subtract $ \dfrac{4197i}{10} $ from $ 6 $ to get $ \dfrac{ \color{purple}{ -4197i+60 } }{ 10 }$. Step 1: Write $ 6 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator. Step 2: To subtract raitonal expressions, both fractions must have the same denominator. |
| ⑦ | Divide $ \dfrac{-416i+3}{3} $ by $ \dfrac{-4197i+60}{10} $ to get $ \dfrac{ -4160i+30 }{ -12591i+180 } $. 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. $$ \begin{aligned} \frac{ \frac{-416i+3}{3} }{ \frac{\color{blue}{-4197i+60}}{\color{blue}{10}} } & \xlongequal{\text{Step 1}} \frac{-416i+3}{3} \cdot \frac{\color{blue}{10}}{\color{blue}{-4197i+60}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( -416i+3 \right) \cdot 10 }{ 3 \cdot \left( -4197i+60 \right) } \xlongequal{\text{Step 3}} \frac{ -4160i+30 }{ -12591i+180 } \end{aligned} $$ |
| ⑧ | Divide $ \, 30-4160i \, $ by $ \, 180-12591i \, $ to get $\,\, \dfrac{5820440-41230i}{17618409} $. ( view steps ) |
| ⑨ | Multiply $5+3i$ by $ \dfrac{5820440-41230i}{17618409} $ to get $ \dfrac{-123690i^2+17255170i+29102200}{17618409} $. Step 1: Write $ 5+3i $ as a fraction by putting $ \color{red}{1} $ in the denominator. Step 2: Multiply numerators and denominators. Step 3: Simplify numerator and denominator. $$ \begin{aligned} 5+3i \cdot \frac{5820440-41230i}{17618409} & \xlongequal{\text{Step 1}} \frac{5+3i}{\color{red}{1}} \cdot \frac{5820440-41230i}{17618409} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( 5+3i \right) \cdot \left( 5820440-41230i \right) }{ 1 \cdot 17618409 } \xlongequal{\text{Step 3}} \frac{ 29102200-206150i+17461320i-123690i^2 }{ 17618409 } = \\[1ex] &= \frac{-123690i^2+17255170i+29102200}{17618409} \end{aligned} $$ |
| ⑩ | $$ -123690i^2 = -123690 \cdot (-1) = 123690 $$ |
| ⑪ | Simplify numerator $$ \color{blue}{123690} +i+ \color{blue}{29102200} = i+ \color{blue}{29225890} $$ |