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* a * ... * a_(mtimes) a^n&=a* a * ... * a_(ntimes)
With this information, the left-hand side of the identity can be rewritten using .
a^m * a^n =
a* a * ... * a_(mtimes) * a* a * ... * a_(ntimes)_(m+ ntimes)
By the definition of a power, multiplying a number by itself m+n times is equivalent to raising that number to the power of m+n.
a* a * ... * a_(mtimes) * a* a * ... * a_(ntimes)_(m+ ntimes) =
a^(m+ n)
The Product of Powers Property has been proved.