Question
Answer
$$i^{119}+i^{42}+5i^{57}-8i^{74}=4i+7$$
Explanation
Tap the blue circles to see an explanation.
| $$ \begin{aligned}i^{119}+i^{42}+5i^{57}-8i^{74}& \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle1}{\textcircled {1}} \htmlClass{explanationCircle explanationCircle2}{\textcircled {2}} \htmlClass{explanationCircle explanationCircle3}{\textcircled {3}} \htmlClass{explanationCircle explanationCircle4}{\textcircled {4}} } }}}-i-1+5i+8 \xlongequal{ } \\[1 em] & \xlongequal{ \color{blue}{ \text{\normalsize{ \htmlClass{explanationCircle explanationCircle5}{\textcircled {5}} } }}}4i+7\end{aligned} $$ | |
| ① | $$ i^{119} = i^{4 \cdot 29 + 3} =
\left( i^4 \right)^{ 29 } \cdot i^3 =
1^{ 29 } \cdot (-i) =
-i = -i $$ |
| ② | $$ i^{42} = i^{4 \cdot 10 + 2} =
\left( i^4 \right)^{ 10 } \cdot i^2 =
1^{ 10 } \cdot (-1) =
-1 = -1 $$ |
| ③ | $$ 5i^{57} = 5 \cdot i^{4 \cdot 14 + 1} =
5 \cdot \left( i^4 \right)^{ 14 } \cdot i^1 =
5 \cdot 1^{ 14 } \cdot i =
5 \cdot i $$ |
| ④ | $$ -8i^{74} = -8 \cdot i^{4 \cdot 18 + 2} =
-8 \cdot \left( i^4 \right)^{ 18 } \cdot i^2 =
-8 \cdot 1^{ 18 } \cdot (-1) =
-8 \cdot -1 = 8 $$ |
| ⑤ | Combine like terms: $$ \color{blue}{-i} + \color{blue}{5i} \color{red}{-1} + \color{red}{8} = \color{blue}{4i} + \color{red}{7} $$ |
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