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: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a^{-n} = \frac{1}{a^{n}} } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a^m)^n = a^{mn} = (a^n)^m }
• Product to a power: $(ab)^{n}=a^{n}b^{n}$
• Quotient to a power: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0^{-n} = 1/0^n = 1/0 } ).
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0^0 } is indeterminate or undefined depending on the context).

Resources

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