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# 虚数单位的幂

## 求解 ${i}^{3}$‍  和 ${i}^{4}$‍

$\begin{array}{rl}{i}^{3}& ={i}^{2}\cdot i\\ \\ & =\left(-1\right)\cdot i\\ \\ & =-i\end{array}$

$\begin{array}{rl}{i}^{4}& ={i}^{2}\cdot {i}^{2}\\ \\ & =\left(-1\right)\cdot \left(-1\right)\\ \\ & =1\end{array}$

## $i$‍  的其他幂

${i}^{1}$${i}^{2}$${i}^{3}$${i}^{4}$${i}^{5}$${i}^{6}$${i}^{7}$${i}^{8}$
$i$$-1$$-i$$1$$i$$-1$$-i$$1$

## 一个新的规律

$i$, $-1$, $-i$, $1$, $i$, $-1$, $-i$, $1$, $i$, $-1$, $-i$, $1$, $i$, $-1$, $-i$, $1$, $i$, $-1$, $-i$, $1$

$\begin{array}{rlrl}{i}^{20}& =\left({i}^{4}{\right)}^{5}& & \text{次方属性}\\ \\ & =\left(1{\right)}^{5}& & {i}^{4}=1\\ \\ & =1& & \text{化简}\end{array}$

## $i$‍  的更高次幂

### 解法

$\begin{array}{rlrl}{i}^{138}& ={i}^{136}\cdot {i}^{2}& & \text{次方属性}\\ \\ & =\left({i}^{4\cdot 34}\right)\cdot {i}^{2}& & 136=4\cdot 34\\ \\ & =\left({i}^{4}{\right)}^{34}\cdot {i}^{2}& & \text{次方属性}\\ \\ & =\left(1{\right)}^{34}\cdot {i}^{2}& & {i}^{4}=1\\ \\ & =1\cdot -1& & {i}^{2}=-1\\ \\ & =-1\end{array}$