Exponential Properties

Introduction

Exponential properties can be used to manipulate equations involving exponential expressions and/or functions. Here are some important exponential properties:

• Negative exponent property: $a^{-n}={\frac {1}{a^{n}}}$ and ${\frac {1}{a^{-n}}}=a^{n}$
• Product of like bases: $a^{m}a^{n}=a^{m+n}$
• Quotient of like bases: ${\frac {a^{m}}{a^{n}}}=a^{m}a^{-n}=a^{m-n}$
• Multiple powers: $(a^{m})^{n}=a^{mn}=(a^{n})^{m}$
• Product to a power: $(ab)^{n}=a^{n}b^{n}$
• Quotient to a power: $\left({\frac {a}{b}}\right)^{n}={\frac {a^{n}}{b^{n}}}$

Special cases involving 0:

• For any nonzero number $a$, $a^{0}=1$.
• For any positive number $m$, $0^{m}=0$.
• $0^{m}$ does not exist if m is negative (since $0^{-n}=1/0^{n}=1/0$).
• $0^{0}$ is indeterminate or undefined depending on the context).

Resources

• Exponential and Logarithmic Properties, Arizona State University
• Exponent Properties Review, Khan Academy