Question
Answer
$$-7i^{26}+i^{74}-2i^{31}-12i^{70}=2i+18$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}-7i^{26}+i^{74}-2i^{31}-12i^{70}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}7-1+2i+12 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}2i+18\end{aligned} $$ | |
| ① | $$ -7i^{26} = -7 \cdot i^{4 \cdot 6 + 2} =
-7 \cdot \left( i^4 \right)^{ 6 } \cdot i^2 =
-7 \cdot 1^{ 6 } \cdot (-1) =
-7 \cdot -1 = 7 $$ |
| ② | $$ i^{74} = i^{4 \cdot 18 + 2} =
\left( i^4 \right)^{ 18 } \cdot i^2 =
1^{ 18 } \cdot (-1) =
-1 = -1 $$ |
| ③ | $$ -2i^{31} = -2 \cdot i^{4 \cdot 7 + 3} =
-2 \cdot \left( i^4 \right)^{ 7 } \cdot i^3 =
-2 \cdot 1^{ 7 } \cdot (-i) =
-2 \cdot -i = 2i $$ |
| ④ | $$ -12i^{70} = -12 \cdot i^{4 \cdot 17 + 2} =
-12 \cdot \left( i^4 \right)^{ 17 } \cdot i^2 =
-12 \cdot 1^{ 17 } \cdot (-1) =
-12 \cdot -1 = 12 $$ |
| ⑤ | Combine like terms: $$ 2i \color{blue}{-1} + \color{red}{7} + \color{red}{12} = 2i+ \color{red}{18} $$ |
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