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幂运算

## 幂运算
在有理数乘方运算中，幂运算是考察的重点，下面这些公式很重要
 $$
 \begin{align}
& a^m\times a^n &=a^{m+n}\\
&  (a\cdot b)^n&=a^n\cdot b^n\\
&  (a^m)^n&=a^{mn}\\
&  (a^0)&=1 \\
&   a^m\div a^n&=a^{m-n}\qquad (a\ne 0,\; m\ge n)\\
& \left(\frac{a}{b}\right)^m&=\frac{a^m}{b^m}\qquad (b\ne 0)
\end{align} 
$$

#### 例题1
$\left[(-2)^m\cdot (-2)^n\right]^2=[(-2)^{m+n}]^2=[(-2)^2]^{m+n}=4^{m+n}$

#### 例题2
$\left[\left(-\frac{1}{2}\right)^m\div \left(-\frac{1}{2}\right)^n\right]^3=\left[\left(-\frac{1}{2}\right)^{m-n}\right]^3=\left[\left(-\frac{1}{2}\right)^3\right]^{m-n}=-\frac{1}{8^{m-n}} $

#### 例题3
$\left(-\frac{1}{3}\right)^{2n}=\left[\left(-\frac{1}{3}\right)^2\right]^n=\left(+\frac{1}{9}\right)^n=+\frac{1}{9^n}  $

#### 例题4
$\left(-\frac{1}{3}\right)^{2n+1}=\left(-\frac{1}{3}\right)^{2n}\cdot \left(-\frac{1}{3}\right)=+\frac{1}{3^{2n}}\cdot \left(-\frac{1}{3}\right)=-\frac{1}{3^{2n+1}}$

#### 例题5
计算：$(-5)^3\cdot (-2)^2\cdot (-5)^2 \cdot \left(+\frac{3}{5}\right)^3$

$$
\begin{align*}
\text{原式}&=[(-2)^2\cdot (-5)^2]\cdot\left[(-5)^3\cdot \left(+\frac{3}{5}\right)^3\right]    \tag{交换律、结合律}\\
&=[(-2)^2\cdot (-5)^2]\cdot\left[(-5)\cdot \frac{3}{5}\right]^3    \tag{指数运算律}\\
&=(+10)^2\cdot (-3)^3 \tag{乘法法则}\\
&=100\times (-27) \tag{乘方运算法则}\\
&=-2700
\end{align*}
$$

#### 例题6
计算$\left[2\frac{1}{4}-\left(-\frac{1}{2}\right)^3\right]\div \left(-\frac{3}{8}\right)+\left(-\frac{3}{5}\right)\cdot \left(-1\frac{2}{3}\right)^2$

解：
$$
 \begin{align*}
        \text{原式} &= \left(\frac{9}{4}+\frac{1}{8}\right)\div \left(-\frac{3}{8}\right)+\left(-\frac{3}{5}\right)\times \frac{25}{9} \tag{先里、乘方}\\
        &= \left(+\frac{19}{8}\right)\times \left(-\frac{8}{3}\right)+\left(-\frac{3}{5}\right)\times \frac{25}{9}\tag{再乘、除}\\
        &= \left(-\frac{19}{3}\right)+\left(-\frac{15}{9}\right)\\
        &= \left(-\frac{19}{3}\right)+\left(-\frac{5}{3}\right)\\
        &= -\left(\frac{19+5}{3}\right)=-8 \tag{后加、减}
    \end{align*}
    $$

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