Lesson 3: Zero as an Exponent

Property

$a^0 = 1$, if $a \neq 0$

This definition is based on the second law of exponents. For any non-zero number $a$, the quotient $\frac{a^n}{a^n}$ is equal to 1. Using the law of exponents, we can also write $\frac{a^n}{a^n} = a^{n-n} = a^0$. Therefore, it is logical to define $a^0$ as 1.

Examples

• For a positive integer, $8^0 = 1$.
• For a negative integer, $(-55)^0 = 1$.
• For an algebraic term where variables are non-zero, $(2ab^2)^0 = 1$.

Property

If $a$ is a non-zero number, then $a^0 = 1$. Any nonzero number raised to the zero power is 1. In this text, we assume any variable that we raise to the zero power is not zero.

Examples

• Any non-zero number to the zero power is 1, so $15^0 = 1$.

• For any non-zero variable $p$, it is true that $p^0 = 1$.