Question
Answer
$$(-3+2i+h)^3=h^3+6h^2i+12hi^2+8i^3-9h^2-36hi-36i^2+27h+54i-27$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}(-3+2i+h)^3& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}h^3+6h^2i+12hi^2+8i^3-9h^2-36hi-36i^2+27h+54i-27\end{aligned} $$ | |
| ① | Multiply each term of $ \left( \color{blue}{-3+2i+h}\right) $ by each term in $ \left( -3+2i+h\right) $. $$ \left( \color{blue}{-3+2i+h}\right) \cdot \left( -3+2i+h\right) = 9-6i-3h-6i+4i^2+2hi-3h+2hi+h^2 $$ |
| ② | Combine like terms: $$ 9 \color{blue}{-6i} \color{red}{-3h} \color{blue}{-6i} +4i^2+ \color{green}{2hi} \color{red}{-3h} + \color{green}{2hi} +h^2 = h^2+ \color{green}{4hi} +4i^2 \color{red}{-6h} \color{blue}{-12i} +9 $$ |
| ③ | Multiply each term of $ \left( \color{blue}{h^2+4hi+4i^2-6h-12i+9}\right) $ by each term in $ \left( -3+2i+h\right) $. $$ \left( \color{blue}{h^2+4hi+4i^2-6h-12i+9}\right) \cdot \left( -3+2i+h\right) = \\ = -3h^2+2h^2i+h^3-12hi+8hi^2+4h^2i-12i^2+8i^3+4hi^2+18h-12hi-6h^2+36i-24i^2-12hi-27+18i+9h $$ |
| ④ | Combine like terms: $$ \color{blue}{-3h^2} + \color{red}{2h^2i} +h^3 \color{green}{-12hi} + \color{orange}{8hi^2} + \color{red}{4h^2i} \color{blue}{-12i^2} +8i^3+ \color{orange}{4hi^2} + \color{red}{18h} \color{green}{-12hi} \color{blue}{-6h^2} + \color{orange}{36i} \color{blue}{-24i^2} \color{green}{-12hi} -27+ \color{orange}{18i} + \color{red}{9h} = \\ = h^3+ \color{red}{6h^2i} + \color{orange}{12hi^2} +8i^3 \color{blue}{-9h^2} \color{green}{-36hi} \color{blue}{-36i^2} + \color{red}{27h} + \color{orange}{54i} -27 $$ |
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