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初级代数

课程: 初级代数 > 单元 6

课程 5: 负指数幂

负指数幂复习

复习负指数幂的基本概念, 并做一些练习题.

负指数幂的定义

我们定义负指数幂是底数的乘法倒数，上升到幂的相反指数。

x, start superscript, minus, n, end superscript, equals, start fraction, 1, divided by, x, start superscript, n, end superscript, end fraction

想要了解更多关于此定义？点击视频。

示例

3, start superscript, minus, 5, end superscript, equals, start fraction, 1, divided by, 3, start superscript, 5, end superscript, end fraction
start fraction, 1, divided by, 2, start superscript, 8, end superscript, end fraction, equals, 2, start superscript, minus, 8, end superscript
y, start superscript, minus, 2, end superscript, equals, start fraction, 1, divided by, y, squared, end fraction
left parenthesis, start fraction, 8, divided by, 6, end fraction, right parenthesis, start superscript, minus, 3, end superscript, equals, left parenthesis, start fraction, 6, divided by, 8, end fraction, right parenthesis, cubed

练习

问题1

当前

选择相等的表达式。

4, start superscript, minus, 3, end superscript, equals, question mark

想要尝试更多此类题目？点击练习。

一些拓展

那么为什么我们要用这种方法来定义负指数幂呢？下面是几个理由:

理由1: 规律

n	2, start superscript, n, end superscript
3	2, cubed, equals, 8
2	2, squared, equals, 4
1	2, start superscript, 1, end superscript, equals, 2
0	2, start superscript, 0, end superscript, equals, 1
minus, 1	2, start superscript, minus, 1, end superscript, equals, start fraction, 1, divided by, 2, end fraction
minus, 2	2, start superscript, minus, 2, end superscript, equals, start fraction, 1, divided by, 4, end fraction

注意到如何2, start superscript, n, end superscript每次我们减少n时都会除以2。此规律即使n为零或者负数仍然持续。

理由2: 指数性质

还记得start fraction, x, start superscript, n, end superscript, divided by, x, start superscript, m, end superscript, end fraction, equals, x, start superscript, n, minus, m, end superscript吗，所以...

\begin{aligned} \dfrac{2^2}{2^3}&=2^{2-3} \\\\ &=2^{-1} \end{aligned}

我们也知道,

\begin{aligned} \dfrac{2^2}{2^3}&=\dfrac{\cancel 2\cdot\cancel 2}{\cancel 2\cdot\cancel 2\cdot 2} \\\\ &=\dfrac12 \end{aligned}

因此, 我们得到 2, start superscript, minus, 1, end superscript, equals, start fraction, 1, divided by, 2, end fraction。

还有，记得x, start superscript, n, end superscript, dot, x, start superscript, m, end superscript, equals, x, start superscript, n, plus, m, end superscript吗，所以...

\begin{aligned} 2^2\cdot 2^{-2}&=2^{2+(-2)} \\\\ &=2^0 \\\\ &=1 \end{aligned}

同理，根据定义...

\begin{aligned} 2^2\cdot 2^{-2}&=2^2\cdot\dfrac{1}{2^2} \\\\ &=\dfrac{2^2}{2^2} \\\\ &=1 \end{aligned}

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