Question
Answer
$$(5+2i)^3=142i+65$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(5+2i)^3& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} } }}}125+150i+60i^2+8i^3 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} } }}}125+150i-60-8i \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}142i+65\end{aligned} $$ | |
| ① | Find $ \left(5+2i\right)^3 $ using formula $$ (A + B) = A^3 + 3A^2B + 3AB^2 + B^3 $$where $ A = 5 $ and $ B = 2i $. $$ \left(5+2i\right)^3 = 5^3+3 \cdot 5^2 \cdot 2i + 3 \cdot 5 \cdot \left( 2i \right)^2+\left( 2i \right)^3 = 125+150i+60i^2+8i^3 $$ |
| ② | $$ 60i^2 = 60 \cdot (-1) = -60 $$ |
| 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: $$ \color{blue}{150i} \color{blue}{-8i} \color{red}{-60} + \color{red}{125} = \color{blue}{142i} + \color{red}{65} $$ |
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