Adding exponents

Algebra

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.