Properties of Powers with Integer Exponents
Big Idea: Exponents are a shorthand for repeated multiplication. When we extend exponents to include all integers (positive, negative, and zero), a beautiful set of consistent rules emerges that makes working with very large and very small numbers elegant and efficient.
Foundation: What is an Integer Exponent?
For any nonzero number $a$ and a positive integer $n$:
$$ a^n = \underbrace{a \cdot a \cdot a \cdots a}_{n \text{ factors}} $$
But what happens when the exponent is zero or negative? We define:
Zero Exponent
$a^0 = 1$
(for $a \neq 0$)
Example: $5^0 = 1$, $(-3)^0 = 1$
Negative Exponent
$a^{-n} = \dfrac{1}{a^n}$
(for $a \neq 0$)
Example: $2^{-3} = \dfrac{1}{2^3} = \dfrac{1}{8}$