Question
Answer
$$-9i^{60}-i^6+5i^{79}+i^{37}=-4i-8$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}-9i^{60}-i^6+5i^{79}+i^{37}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-9+1-5i+i \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-4i-8\end{aligned} $$ | |
| ① | $$ -9i^{60} = -9 \cdot i^{4 \cdot 15 + 0} =
-9 \cdot \left( i^4 \right)^{ 15 } \cdot i^0 =
-9 \cdot 1^{ 15 } \cdot 1 =
-9 \cdot 1 $$ |
| ② | $$ -i^6 = - i^{4 \cdot 1 + 2} =
- \left( i^4 \right)^{ 1 } \cdot i^2 =
- 1^{ 1 } \cdot (-1) =
- -1 = 1 $$ |
| ③ | $$ 5i^{79} = 5 \cdot i^{4 \cdot 19 + 3} =
5 \cdot \left( i^4 \right)^{ 19 } \cdot i^3 =
5 \cdot 1^{ 19 } \cdot (-i) =
5 \cdot -i = -5i $$ |
| ④ | $$ i^{37} = i^{4 \cdot 9 + 1} =
\left( i^4 \right)^{ 9 } \cdot i^1 =
1^{ 9 } \cdot i =
i $$ |
| ⑤ | Combine like terms: $$ \color{blue}{-5i} + \color{blue}{i} \color{red}{-9} + \color{red}{1} = \color{blue}{-4i} \color{red}{-8} $$ |
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