指数运算和幂运算

1. Exponentiation (指数运算和幂运算)

In mathematics, exponentiation is an operation involving two numbers: the base and the exponent or power. Exponentiation is written as $b^n$, where $b$ is the base and $n$ is the power; this is pronounced as “$b$ (raised) to the (power of) $n$”.
在数学中，重复连乘的运算叫做乘方，乘方的结果称为幂。在 $b^n$ 中，底数为 $b$，指数为 $n$。

When $n$ is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, $b^n$ is the product of multiplying $n$ bases.
若 $n$ 为正整数，$n$ 个相同的数 $b$ 连续相乘 (即 $b$ 自乘 $n$ 次)，就可将 $b^n$ 看作乘方的结果幂。
$$b^n=\underset{n\text{ times}}{\underbrace{b\times b\times \cdots \times b\times b}}$$

In particular, $b^1=b$.
当指数为 1 时，通常不写出来，因为运算出的值和底数的数值一样。

The expression $b^2=b\ast b$ is called “the square of $b$” or “$b$ squared”, because the area of a square with side-length $b$ is $b^2$.
指数为 2 时，可以读作 $b$ 的平方。

the expression $b^3=b\ast b\ast b$ is called “the cube of $b$” or “$b$ cubed”, because the volume of a cube with side-length $b$ is $b^3$.
指数为 3 时，可以读作 $b$ 的立方。

mathematics /ˌmæθəˈmætɪks/：n. 数学，计算，运算
exponentiation /ˌekspoʊˌnenʃɪ'eɪʃən/：n. 指数运算，幂运算
exponent /ɪkˈspəʊnənt/：n. 指数，幂，(观点、理论的) 拥护者，鼓吹者，倡导者，(某种活动的) 能手，大师 adj. 讲解的
power /ˈpaʊə(r)/：n. 权力，能力，操纵力，职权，政权，体力，强国，实力，影响力，动力，能量，电力供应，幂，放大率，势力 v. 驱动，快速前进 adj. 电动的

The exponent is usually shown as a superscript to the right of the base as $b^n$ or in computer code as b^n. In that case, $b^n$ is called “$b$ raised to the $n$th power”, “$b$ to the power of $n$”, “the $n$th power of $b$”, or most briefly as “$b$ to the $n$”.
幂运算 (exponentiation) 又称指数运算，表达式为 $b^n$，读作 $b$ 的 $n$ 次方或 $b$ 的 $n$ 次幂。其中 $b$ 称为底数，而 $n$ 称为指数，通常指数写成上标，放在底数的右边。$b^n$ 通常写成 b^n 或 b**n。

The above definition of $b^n$ immediately implies several properties, in particular the multiplication rule:

$$b^n\times b^m=\underset{n\text{ times}}{\underbrace{b\times \cdots \times b}}\times \underset{m\text{ times}}{\underbrace{b\times \cdots \times b}}=\underset{n+m\text{ times}}{\underbrace{b\times \cdots \times b}}=b^{n+m}$$

That is, when multiplying a base raised to one power times the same base raised to another power, the powers add.

Extending this rule to the power zero gives $b^0\times b^n=b^{0+n}=b^n$, and dividing both sides by $b^n$ gives $b^0=b^n/b^n=1$. That is, the multiplication rule implies the definition $b^0=1$.
指数是 0 时，底数不为 0，幂均为 1 (即除 0 外，所有数的 0 次方都是 1)：
$$b^0=1(b\ne 0)$$
0 的 0 次方 ($0^0$) 目前数学家没有给予正式的定义，在部分数学领域中，常用的惯例是定义为 1 。

A similar argument implies the definition for negative integer powers: $b^{-n}=1/b^n$.
指数是负数时，就等于重复除以底数 (或底数的倒数自乘指数这么多次)：
$$b^{-n}=\frac{1}{\underset{n}{\underbrace{b\times \cdots \times b}}}=\frac{1}{b^n}={\left(\frac{1}{b}\right)}^n(b\ne 0)$$

That is, extending the multiplication rule gives $b^{-n}\times b^n=b^{-n+n}=b^0=1$. Dividing both sides by $b^n$ gives $b^{-n}=1/b^n$. This also implies the definition for fractional powers: $b^{n/m}=\sqrt[m]{b^n}$.
若以分数为指数的幂，则定义 $b^{\frac{n}{m}}=\sqrt[m]{b^n}$，即 $b$ 的 $n$ 次方再开 $m$ 次方根。

For example, $b^{1/2}\times b^{1/2}=b^{1/2+1/2}=b^1=b$, meaning $(b^{1/2}{)}^2=b$, which is the definition of square root: $b^{1/2}=\sqrt{b}$.

2. Exponent rules (指数定律)

• $a^m\times a^n=a^{m+n}$

同底数幂相乘，底数不变，指数相加。

• $a^m÷a^n=a^{m-n}$

同底数幂相除，底数不变，指数相减。

• $\frac{a^n}{b^n}={\left(\frac{a}{b}\right)}^n$

同指数幂相除，指数不变，底数相除 ($b$ 不为 0)。

• $a^n\cdot b^n=(a\cdot b{)}^n$

• ${\left(a^m\right)}^n=a^{m\cdot n}$

• $x^{\frac{m}{n}}=\sqrt[n]{x^m}$

• $x^{-m}=\frac{1}{x^m}(x\ne 0)$

• $x^0=1(x\ne 0)$

• $x^1=x$

• $x^{-1}=\frac{1}{x}(x\ne 0)$

3. Particular bases (特殊底数的幂)

• Powers of ten (10 的幂)

In the base ten (decimal) number system, integer powers of 10 are written as the digit 1 followed or preceded by a number of zeroes determined by the sign and magnitude of the exponent. For example, $10^3=1000$, $10^{-3}=0.001$ and $10^{-4}=0.0001$.
在十进制的计数系统中，10 的幂写成 1 后面跟着很多个 0。

• Powers of two (2 的幂)

The first negative powers of 2 have special names: $2^{-1}$ is a half; $2^{-2}$ is a quarter.

• Powers of one (1 的幂)

Every power of one equals: $1^n=1$.
1 的任何次幂都为 1。

• Powers of zero (0 的幂)

For a positive exponent $n>0$, the $n$th power of zero is zero: $0^n=0$. For a negative exponent, $0^{-n}=1/0^n=1/0$ is undefined.
0 的正数幂都等于 0，0 的负数幂没有定义。

The expression $0^0$ is either defined as $\underset{x\to 0}{\lim }{x}^x=1$, or it is left undefined.
任何非 0 之数的 0 次方都是 1；而 0 的 0 次方是悬而未决的，某些领域下常用的惯例是约定为 1。

• Powers of negative one (负 1 的幂)

Since a negative number times another negative is positive, we have:
$$(-1{)}^n=\{\begin{array}{rl}1& \text{for even }n,\\ -1& \text{for odd }n.\end{array}$$

Because of this, powers of −1 are useful for expressing alternating sequences.
-1 的奇数幂等于 -1，-1 的偶数幂等于 1。

• Large exponents (指数非常大时的幂)

The limit of a sequence of powers of a number greater than one diverges; in other words, the sequence grows without bound: $b^n\to \infty$ as $n\to \infty$ when $b>1$
This can be read as “$b$ to the power of $n$ tends to $+\infty$ as $n$ tends to infinity when $b$ is greater than one”.
一个大于 1 的数的幂趋于无穷大，一个小于 -1 的数的幂趋于负无穷大。
当 $a>1$，$n\to \infty$，$a^n\to \infty$
当 $a<-1$，$n\to \infty$，$a^n\to -\infty$ 或 $\infty$ (取决于 $n$ 是奇数或偶数)

Powers of a number with absolute value less than one tend to zero: $b^n\to 0$ as $n\to \infty$ when $|b|<1$
一个绝对值小于 1 的数的幂趋于 0，当 $|a|<1$，$n\to \infty$，$a^n\to 0$

Any power of one is always one: $b^n=1$ for all $n$ for $b=1$
1 的幂永远都是 1，当 $a=1$，$n\to \infty$，$a^n\to 1$

Powers of a negative number $b\le -1$ alternate between positive and negative as $n$ alternates between even and odd, and thus do not tend to any limit as $n$ grows.
当 $n\to \infty$, ${\left(1+\frac{1}{n}\right)}^n\to e$

4. Integer exponents (整数指数的幂)

• Positive exponents (正整数指数的幂)

The base case is $b^1=b$ and the recurrence is $b^{n+1}=b^n\cdot b$.

The associativity of multiplication implies that for any positive integers $m$ and $n$, $b^{m+n}=b^m\cdot b^n$, and ${\left(b^m\right)}^n=b^{m\cdot n}$.

• Zero exponent (指数 0 的幂)

As mentioned earlier, a (nonzero) number raised to the 0 power is 1: $b^0=1$.

• Negative exponents (负数指数的幂)

Exponentiation with negative exponents is defined by the following identity, which holds for any integer $n$ and nonzero $b$: $b^{-n}=\frac{1}{b^n}$.
任何不为 0 的数 a 的 -1 次方等于它的倒数 $a^{-1}=\frac{1}{a}$。

对于非零 $a$ 定义 $a^{-n}=\frac{1}{a^n}$，而 $a=0$ 时分母为 0 没有意义。

根据定义 $a^m\cdot a^n=a^{m+n}$，当 $m=-n$ 时 $a^{-n}{a}^n=a^{-n+n}=a^0=1$，得 $a^{-n}{a}^n=1$, 所以 $a^{-n}=\frac{1}{a^n}$。

通过运算法则 $\frac{a^m}{a^n}=a^{m-n}$，当 $m=0$ 时，可得 $a^{-n}=a^{0-n}=\frac{a^0}{a^n}=\frac{1}{a^n}$

References

[1] Yongqiang Cheng, https://yongqiang.blog.csdn.net/
[2] Exponentiation, https://en.wikipedia.org/wiki/Exponentiation
[3] Exponent rules, https://www.rapidtables.com/math/number/exponent.html
[4] Exponential function, https://en.wikipedia.org/wiki/Exponential_function