### Rule for Exponents and Negative Exponents

**Exponents** are a shorthand way to represent repeated multiplication of a number by itself. For a base $a$ raised to the power of $n$, we write $a^n$, which means:

$a^n = \underbrace{a \times a \times \cdots \times a}_{n \text{ times}}$
### Positive Exponents

For positive exponents, $a^3$ means multiplying the base $a$ three times:

$a^3 = a \times a \times a$
### Zero Exponent

Any non-zero base raised to the power of zero is always 1:

$a^0 = 1 \quad \text{for } a \neq 0$
### Negative Exponents

Negative exponents represent the reciprocal of the base raised to the absolute value of the exponent. For a base $a$ raised to the power of $-n$, we write $a^{-n}$, which means:

$a^{-n} = \frac{1}{a^n}$
For example:

$a^{-2} = \frac{1}{a^2}$
### Example Calculation

Let’s calculate some examples to illustrate these rules:

- $2^3 = 2 \times 2 \times 2 = 8$
- $5^0 = 1$
- $3^{-2} = \frac{1}{3^2} = \frac{1}{9}$

### Combining Exponents

When multiplying or dividing like bases, you can combine the exponents:

**Multiplication**: $a^m \times a^n = a^{m+n}$

$2^3 \times 2^2 = 2^{3+2} = 2^5 = 32$
**Division**: $\frac{a^m}{a^n} = a^{m-n}$

$\frac{3^4}{3^2} = 3^{4-2} = 3^2 = 9$
Understanding these rules allows you to simplify and compute expressions involving exponents and negative exponents efficiently.