Question
Answer
$$-8i^2-9i^{42}+i^{83}+2i^{62}=-i+15$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}-8i^2-9i^{42}+i^{83}+2i^{62}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}8+9-i-2 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-i+15\end{aligned} $$ | |
| ① | $$ -8i^2 = -8 \cdot (-1) = 8 $$ |
| ② | $$ -9i^{42} = -9 \cdot i^{4 \cdot 10 + 2} =
-9 \cdot \left( i^4 \right)^{ 10 } \cdot i^2 =
-9 \cdot 1^{ 10 } \cdot (-1) =
-9 \cdot -1 = 9 $$ |
| ③ | $$ i^{83} = i^{4 \cdot 20 + 3} =
\left( i^4 \right)^{ 20 } \cdot i^3 =
1^{ 20 } \cdot (-i) =
-i = -i $$ |
| ④ | $$ 2i^{62} = 2 \cdot i^{4 \cdot 15 + 2} =
2 \cdot \left( i^4 \right)^{ 15 } \cdot i^2 =
2 \cdot 1^{ 15 } \cdot (-1) =
2 \cdot -1 = -2 $$ |
| ⑤ | Combine like terms: $$ -i+ \color{blue}{9} + \color{red}{8} \color{red}{-2} = -i+ \color{red}{15} $$ |
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