# Exponent Combination Laws/Sum of Powers

 It has been suggested that this article or section be renamed: $a^x + a^y$ would be a sum of powers. Perhaps easiest to resolve as "Product of Powers" or "Power with Sum Exponent" One may discuss this suggestion on the talk page.

## Theorem

Let $a \in \R_{> 0}$ be a positive real number.

Let $x, y \in \R$ be real numbers.

Let $a^x$ be defined as $a$ to the power of $x$.

Then:

$a^{x + y} = a^x a^y$

## Proof

 $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle a^{x + y}$$ $$=$$ $$\displaystyle$$ $$\displaystyle \exp \left({\left({x + y}\right) \ln a}\right)$$ $$\displaystyle$$ $$\displaystyle$$ Definition of Power to Real Number $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$=$$ $$\displaystyle$$ $$\displaystyle \exp \left({x \ln a + y \ln a}\right)$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$=$$ $$\displaystyle$$ $$\displaystyle \exp \left({x \ln a}\right) \exp \left({y \ln a}\right)$$ $$\displaystyle$$ $$\displaystyle$$ Exponent of Sum $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$\displaystyle$$ $$=$$ $$\displaystyle$$ $$\displaystyle a^x a^y$$ $$\displaystyle$$ $$\displaystyle$$ Definition of Power to Real Number

$\blacksquare$