## Statement

When multiplying powers with the same base, you can add the exponents.

$$ a^b * a^c = a^{b + c} $$

## Proof

From the definition of powers, write $ a^b * a^c $ as factors and combine them to a new power with exponent $ b + c $.

$$ a^b * a^c = \underbrace{a * a * a * ...}_\text{b times} * \underbrace{a * a * a * a * ...}_\text{c times} = a^{b + c} $$