### Product of Powers

When you multiply numbers with the same base, you add their exponents. If $a$ is any non-zero number and $m$ and $n$ are integers, then:

$a^m \cdot a^n = a^{m+n}$
### Quotient of Powers

When you divide numbers with the same base, you subtract the exponent of the denominator from the exponent of the numerator:

$\frac{a^m}{a^n} = a^{m-n} \quad \text{for } a \neq 0$
### Power of a Power

When raising an exponent to another exponent, you multiply the exponents:

$(a^m)^n = a^{m \cdot n}$
### Power of a Product

When raising a product to an exponent, you distribute the exponent to both bases:

$(ab)^m = a^m \cdot b^m$
### Power of a Quotient

When raising a quotient to an exponent, you distribute the exponent to both the numerator and the denominator:

$\left( \frac{a}{b} \right)^m = \frac{a^m}{b^m}$
### Zero Exponent

Any non-zero base raised to the zero power is equal to one:

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

A negative exponent indicates the reciprocal of the base raised to the opposite positive exponent:

$a^{-m} = \frac{1}{a^m} \quad \text{for } a \neq 0$
### Fractional Exponent

A fractional exponent indicates both a root and a power. Specifically, if the exponent is $\frac{m}{n}$, it means raise the base to the power of $m$ and then take the $n$th root:

$a^{\frac{m}{n}} = \sqrt[n]{a^m}$
Understanding these **properties of exponents** can greatly help in simplifying expressions and solving equations that involve exponents.