Question
Answer
$$\dfrac{(3+2i)^3}{3-i}=\dfrac{-73+129i}{10}$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{(3+2i)^3}{3-i}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{27+54i+36i^2+8i^3}{3-i} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} } }}}\frac{27+54i-36-8i}{3-i} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{46i-9}{3-i} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{-73+129i}{10}\end{aligned} $$ | |
| ① | Find $ \left(3+2i\right)^3 $ using formula $$ (A + B) = A^3 + 3A^2B + 3AB^2 + B^3 $$where $ A = 3 $ and $ B = 2i $. $$ \left(3+2i\right)^3 = 3^3+3 \cdot 3^2 \cdot 2i + 3 \cdot 3 \cdot \left( 2i \right)^2+\left( 2i \right)^3 = 27+54i+36i^2+8i^3 $$ |
| ② | $$ 36i^2 = 36 \cdot (-1) = -36 $$$$ 8i^3 = 8 \cdot \color{blue}{i^2} \cdot i =
8 \cdot ( \color{blue}{-1}) \cdot i =
-8 \cdot \, i $$ |
| ③ | Simplify numerator $$ \color{blue}{27} + \color{red}{54i} \color{blue}{-36} \color{red}{-8i} = \color{red}{46i} \color{blue}{-9} $$ |
| ④ | Divide $ \, -9+46i \, $ by $ \, 3-i \, $ to get $\,\, \dfrac{-73+129i}{10} $. ( view steps ) |
This page was created using
Complex Expression calculator