Question
Answer
$$11i^{59}+6i^{12}+9i^{15}+i^{11}=-21i+6$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}11i^{59}+6i^{12}+9i^{15}+i^{11}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-11i+6-9i-i \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}-21i+6\end{aligned} $$ | |
| ① | $$ 11i^{59} = 11 \cdot i^{4 \cdot 14 + 3} =
11 \cdot \left( i^4 \right)^{ 14 } \cdot i^3 =
11 \cdot 1^{ 14 } \cdot (-i) =
11 \cdot -i = -11i $$ |
| ② | $$ 6i^{12} = 6 \cdot i^{4 \cdot 3 + 0} =
6 \cdot \left( i^4 \right)^{ 3 } \cdot i^0 =
6 \cdot 1^{ 3 } \cdot 1 =
6 \cdot 1 $$ |
| ③ | $$ 9i^{15} = 9 \cdot i^{4 \cdot 3 + 3} =
9 \cdot \left( i^4 \right)^{ 3 } \cdot i^3 =
9 \cdot 1^{ 3 } \cdot (-i) =
9 \cdot -i = -9i $$ |
| ④ | $$ i^{11} = i^{4 \cdot 2 + 3} =
\left( i^4 \right)^{ 2 } \cdot i^3 =
1^{ 2 } \cdot (-i) =
-i = -i $$ |
| ⑤ | Combine like terms: $$ \color{blue}{-11i} \color{red}{-9i} \color{red}{-i} +6 = \color{red}{-21i} +6 $$ |
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