Question
Answer
$$i^{44}+i^{150}-i^{74}-i^{109}+i^{61}=1$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}i^{44}+i^{150}-i^{74}-i^{109}+i^{61}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}1-1+1-i+i \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle6}{\textcircled {6}} } }}}1\end{aligned} $$ | |
| ① | $$ i^{44} = i^{4 \cdot 11 + 0} =
\left( i^4 \right)^{ 11 } \cdot i^0 =
1^{ 11 } \cdot 1 =
1 $$ |
| ② | $$ i^{150} = i^{4 \cdot 37 + 2} =
\left( i^4 \right)^{ 37 } \cdot i^2 =
1^{ 37 } \cdot (-1) =
-1 = -1 $$ |
| ③ | $$ -i^{74} = - i^{4 \cdot 18 + 2} =
- \left( i^4 \right)^{ 18 } \cdot i^2 =
- 1^{ 18 } \cdot (-1) =
- -1 = 1 $$ |
| ④ | $$ -i^{109} = - i^{4 \cdot 27 + 1} =
- \left( i^4 \right)^{ 27 } \cdot i^1 =
- 1^{ 27 } \cdot i =
- i $$ |
| ⑤ | $$ i^{61} = i^{4 \cdot 15 + 1} =
\left( i^4 \right)^{ 15 } \cdot i^1 =
1^{ 15 } \cdot i =
i $$ |
| ⑥ | Combine like terms: $$ \, \color{blue}{ -\cancel{i}} \,+ \, \color{blue}{ \cancel{i}} \,+ \, \color{green}{ \cancel{1}} \,+ \, \color{blue}{ \cancel{1}} \, \, \color{blue}{ -\cancel{1}} \, = \color{blue}{1} $$ |
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