Properties of powers

From the definition of a power follow five calculation rules that we will use constantly to simplify expressions.

Property — of powers

Let $a,b\in\mathbb{R}\setminus\{0\}$ and $m,n\in\mathbb{Z}$:

Equality	Name
$a^m \cdot a^n = a^{m+n}$	product of powers with the same base
$\dfrac{a^m}{a^n} = a^{m-n}$	quotient of powers with the same base
$(a^m)^n = a^{m \cdot n}$	power of a power
$(a\cdot b)^n = a^n \cdot b^n$	power of a product
$\left(\dfrac{a}{b}\right)^n = \dfrac{a^n}{b^n}$	power of a quotient

The first three rules concern powers with the same base; the last two concern powers with the same exponent.

Links

Topics: Numbers and operations
Concepts: Power · Properties of powers
Skills: Simplifying · Using formulae