Question
Answer
$$\dfrac{(3-79i)^2}{2(11+2i)^3}=i-2$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}\frac{(3-79i)^2}{2(11+2i)^3}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}\frac{(3-79i)^2}{2(1331+726i+132i^2+8i^3)} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}\frac{(3-79i)^2}{2(1331+726i-132-8i)} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}\frac{(3-79i)^2}{2(718i+1199)} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}\frac{9-474i+6241i^2}{1436i+2398} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle7}{\textcircled {7}} } }}}\frac{9-474i-6241}{1436i+2398} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle8}{\textcircled {8}} } }}}\frac{-474i-6232}{1436i+2398} \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle9}{\textcircled {9}} } }}}-2+i \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle10}{\textcircled {10}} } }}}i-2\end{aligned} $$ | |
| ① | Find $ \left(11+2i\right)^3 $ using formula $$ (A + B) = A^3 + 3A^2B + 3AB^2 + B^3 $$where $ A = 11 $ and $ B = 2i $. $$ \left(11+2i\right)^3 = 11^3+3 \cdot 11^2 \cdot 2i + 3 \cdot 11 \cdot \left( 2i \right)^2+\left( 2i \right)^3 = 1331+726i+132i^2+8i^3 $$ |
| ② | $$ 132i^2 = 132 \cdot (-1) = -132 $$ |
| ③ | $$ 8i^3 = 8 \cdot \color{blue}{i^2} \cdot i =
8 \cdot ( \color{blue}{-1}) \cdot i =
-8 \cdot \, i $$ |
| ④ | Combine like terms: $$ \color{blue}{1331} + \color{red}{726i} \color{blue}{-132} \color{red}{-8i} = \color{red}{718i} + \color{blue}{1199} $$ |
| ⑤ | Find $ \left(3-79i\right)^2 $ using formula. $$ (A - B)^2 = \color{blue}{A^2} - 2 \cdot A \cdot B + \color{red}{B^2} $$where $ A = \color{blue}{ 3 } $ and $ B = \color{red}{ 79i }$. $$ \begin{aligned}\left(3-79i\right)^2 = \color{blue}{3^2} -2 \cdot 3 \cdot 79i + \color{red}{\left( 79i \right)^2} = 9-474i+6241i^2\end{aligned} $$ |
| ⑥ | Multiply $ \color{blue}{2} $ by $ \left( 718i+1199\right) $ $$ \color{blue}{2} \cdot \left( 718i+1199\right) = 1436i+2398 $$ |
| ⑦ | $$ 6241i^2 = 6241 \cdot (-1) = -6241 $$ |
| ⑧ | Simplify numerator $$ \color{blue}{9} -474i \color{blue}{-6241} = -474i \color{blue}{-6232} $$ |
| ⑨ | Divide $ \, -6232-474i \, $ by $ \, 2398+1436i \, $ to get $\,\, -2+i $. ( view steps ) |
| ⑩ | Combine like terms: $$ i-2 = i-2 $$ |
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