Question
Answer
$$8i^7+4i^{37}-11i^{68}+11i^{41}=7i-11$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}8i^7+4i^{37}-11i^{68}+11i^{41}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-8i+4i-11+11i \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}7i-11\end{aligned} $$ | |
| ① | $$ 8i^7 = 8 \cdot i^{4 \cdot 1 + 3} =
8 \cdot \left( i^4 \right)^{ 1 } \cdot i^3 =
8 \cdot 1^{ 1 } \cdot (-i) =
8 \cdot -i = -8i $$ |
| ② | $$ 4i^{37} = 4 \cdot i^{4 \cdot 9 + 1} =
4 \cdot \left( i^4 \right)^{ 9 } \cdot i^1 =
4 \cdot 1^{ 9 } \cdot i =
4 \cdot i $$ |
| ③ | $$ -11i^{68} = -11 \cdot i^{4 \cdot 17 + 0} =
-11 \cdot \left( i^4 \right)^{ 17 } \cdot i^0 =
-11 \cdot 1^{ 17 } \cdot 1 =
-11 \cdot 1 $$ |
| ④ | $$ 11i^{41} = 11 \cdot i^{4 \cdot 10 + 1} =
11 \cdot \left( i^4 \right)^{ 10 } \cdot i^1 =
11 \cdot 1^{ 10 } \cdot i =
11 \cdot i $$ |
| ⑤ | Combine like terms: $$ \color{blue}{-8i} + \color{red}{4i} + \color{red}{11i} -11 = \color{red}{7i} -11 $$ |
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