指数规则
指数规则，指数的法则和示例。
什么是指数
基数 a 的 n 次幂等于 a 乘以自身 n 次：
a n = a × a × ... × a
n 次
a 是基数，n 是指数。
示例
31 = 3
32 = 3 × 3 = 9
33 = 3 × 3 × 3 = 27
34 = 3 × 3 × 3 × 3 = 81
35 = 3 × 3 × 3 × 3 × 3 = 243
指数规则和性质
| 规则名称 | 规则 | 示例 |
|---|---|---|
| 乘法规则 | a n ⋅ a m = a n+m | 23 ⋅ 24 = 23+4 = 128 |
| a n ⋅ b n = (a ⋅ b) n | 32 ⋅ 42 = (3⋅4)2 = 144 | |
| 除法规则 | a n / a m = a n-m | 25 / 23 = 25-3 = 4 |
| a n / b n = (a / b) n | 43 / 23 = (4/2)3 = 8 | |
| 幂次规则 | (bn)m = bn⋅m | (23)2 = 23⋅2 = 64 |
| bnm = b(nm) | 232 = 2(32)= 512 | |
| m√(bn) = b < i>n/m | 2√(26) = 26/2 = 8 | |
| b1/n = n√b | 81/3 = 3√8 = 2 | |
| 负指数 | b-n = 1 / bn | 2-3 = 1/23 = 0.125 |
| 零规则 | b0 = 1 | 50 = 1 |
| 0n = 0 , 对于 n>0 | 05 = 0 | |
| 一规则 | b1 = b | 51 = 5 |
| 1n = 1 | 15 = 1 | |
| 负一规则 |  |
(-1)5 = -1 |
| 导数规则 | (xn)' = n⋅x n-1 | (x3)' = 3⋅x3-1 |
| 积分规则 | ∫ xndx = xn+1/(n+1)+C | ∫ x2dx = x2+1/(2+1)+C |
指数乘法规则
相同底数的乘法规则
an ⋅ am = an+m
示例：
23 ⋅ 24 = 23+4 = 27 = 2⋅2⋅2⋅2⋅2⋅2⋅2 = 128
相同指数的乘法规则
an ⋅ bn = (a ⋅ b)n
示例：
32 ⋅ 42 = (3⋅4)2 = 122 = 12⋅12 = 144
查看：指数相乘
指数除法规则
相同底数的除法规则
an / am = an-m
示例：
25 / 23 = 25-3 = 22 = 2⋅2 = 4
相同指数的除法规则
an / bn = (a / b)n
示例：
43 / 23 = (4/2)3 = 23 = 2⋅2⋅2 = 8
查看：指数相除
指数幂运算规则
幂运算规则 I
(an) m = a n⋅m
示例：
(23)2 = 23⋅2 = 26 = 2⋅2⋅2⋅2⋅2⋅2 = 64
幂运算规则 II
a nm = a (nm)
示例：
232 = 2(32) = 2(3⋅3) = 29 = 2⋅2⋅2⋅2⋅2⋅2⋅2⋅2⋅2 = 512
幂运算规则与根式
m√(a n) = a n/m
示例：
2√(26) = 26/2 = 23 = 2⋅2⋅2 = 8
负指数规则
b-n = 1 / bn
示例：
2-3 = 1/23 = 1/(2⋅2⋅2) = 1/8 = 0.125
查看：负指数