Quotient of Powers Property

To prove the identity, it will be shown that the left-hand side is equivalent to the right-hand side. To do so, a^m and a^n will be written as products using the definition of a power. a^m/a^n =a * a * a * ... * a^(m times)/a * a * a * ... * a_(n times) Next, the common factors will be eliminated. It will be arbitrarily assumed that m>n. a^m/a^n =a * a * a * ... * a^(m times)/a * a * a * ... * a_(n times) Note that when the quotient is simplified, the number of remaining factors will be m-n. a^m/a^n =a* a * ... * a_(m-ntimes) Finally, the definition of a power will be used one more time to express the right-hand side of the above equation as a single power. With this last step, the Quotient of Powers Property is proven.

Note that this property is also valid for m≤ n. If m=n, then the exponent will be zero. If mnegative.