Question
Answer
$$-8i^{77}+2i^{34}+7i^{97}+8i^{100}=-i+6$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}-8i^{77}+2i^{34}+7i^{97}+8i^{100}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-8i-2+7i+8 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-i+6\end{aligned} $$ | |
| ① | $$ -8i^{77} = -8 \cdot i^{4 \cdot 19 + 1} =
-8 \cdot \left( i^4 \right)^{ 19 } \cdot i^1 =
-8 \cdot 1^{ 19 } \cdot i =
-8 \cdot i $$ |
| ② | $$ 2i^{34} = 2 \cdot i^{4 \cdot 8 + 2} =
2 \cdot \left( i^4 \right)^{ 8 } \cdot i^2 =
2 \cdot 1^{ 8 } \cdot (-1) =
2 \cdot -1 = -2 $$ |
| ③ | $$ 7i^{97} = 7 \cdot i^{4 \cdot 24 + 1} =
7 \cdot \left( i^4 \right)^{ 24 } \cdot i^1 =
7 \cdot 1^{ 24 } \cdot i =
7 \cdot i $$ |
| ④ | $$ 8i^{100} = 8 \cdot i^{4 \cdot 25 + 0} =
8 \cdot \left( i^4 \right)^{ 25 } \cdot i^0 =
8 \cdot 1^{ 25 } \cdot 1 =
8 \cdot 1 $$ |
| ⑤ | Combine like terms: $$ \color{blue}{-8i} + \color{blue}{7i} \color{red}{-2} + \color{red}{8} = \color{blue}{-i} + \color{red}{6} $$ |
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