The product of two powers with the same non-zero base a and integer exponents m and n can be written as a single power with base a and exponentm+n.
am⋅an=am+n
Proof
To prove the identity, it will be shown that the left-hand side is equivalent to the right-hand side. To do so, am and an will be written as products using the definition of a power.
aman=mtimesa⋅a⋅…⋅a=ntimesa⋅a⋅…⋅a
With this information, the left-hand side of the identity can be rewritten using multiplication.
am⋅an=m+ntimesmtimesa⋅a⋅…⋅a⋅ntimesa⋅a⋅…⋅a
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.