Question
Answer
$$2i^{68}+i^{71}+3i^{40}+6i^9=5i+5$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}2i^{68}+i^{71}+3i^{40}+6i^9& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}2-i+3+6i \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}5i+5\end{aligned} $$ | |
| ① | $$ 2i^{68} = 2 \cdot i^{4 \cdot 17 + 0} =
2 \cdot \left( i^4 \right)^{ 17 } \cdot i^0 =
2 \cdot 1^{ 17 } \cdot 1 =
2 \cdot 1 $$ |
| ② | $$ i^{71} = i^{4 \cdot 17 + 3} =
\left( i^4 \right)^{ 17 } \cdot i^3 =
1^{ 17 } \cdot (-i) =
-i = -i $$ |
| ③ | $$ 3i^{40} = 3 \cdot i^{4 \cdot 10 + 0} =
3 \cdot \left( i^4 \right)^{ 10 } \cdot i^0 =
3 \cdot 1^{ 10 } \cdot 1 =
3 \cdot 1 $$ |
| ④ | $$ 6i^9 = 6 \cdot i^{4 \cdot 2 + 1} =
6 \cdot \left( i^4 \right)^{ 2 } \cdot i^1 =
6 \cdot 1^{ 2 } \cdot i =
6 \cdot i $$ |
| ⑤ | Combine like terms: $$ \color{blue}{-i} + \color{blue}{6i} + \color{red}{3} + \color{red}{2} = \color{blue}{5i} + \color{red}{5} $$ |
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