## 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 $b + c$.

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

## Proofs building upon this proof

### Power rule in calculus

This proofs shows that the derivative of x^r is r * x^(r - 1).

### The derivative of e to the x

This proof shows that the definition of e^x is e^x.

### The sum of logarithms

This proof shows that the sum of two logarithms with the same base is just one logarithm with the inside parts being multiplied.