Question
Answer
$$8i^{119}+i^{78}-3i^{97}+9i^{44}=-11i+8$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}8i^{119}+i^{78}-3i^{97}+9i^{44}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-8i-1-3i+9 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-11i+8\end{aligned} $$ | |
| ① | $$ 8i^{119} = 8 \cdot i^{4 \cdot 29 + 3} =
8 \cdot \left( i^4 \right)^{ 29 } \cdot i^3 =
8 \cdot 1^{ 29 } \cdot (-i) =
8 \cdot -i = -8i $$ |
| ② | $$ i^{78} = i^{4 \cdot 19 + 2} =
\left( i^4 \right)^{ 19 } \cdot i^2 =
1^{ 19 } \cdot (-1) =
-1 = -1 $$ |
| ③ | $$ -3i^{97} = -3 \cdot i^{4 \cdot 24 + 1} =
-3 \cdot \left( i^4 \right)^{ 24 } \cdot i^1 =
-3 \cdot 1^{ 24 } \cdot i =
-3 \cdot i $$ |
| ④ | $$ 9i^{44} = 9 \cdot i^{4 \cdot 11 + 0} =
9 \cdot \left( i^4 \right)^{ 11 } \cdot i^0 =
9 \cdot 1^{ 11 } \cdot 1 =
9 \cdot 1 $$ |
| ⑤ | Combine like terms: $$ \color{blue}{-8i} \color{blue}{-3i} \color{red}{-1} + \color{red}{9} = \color{blue}{-11i} + \color{red}{8} $$ |
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