# Multiplying exponents

## Statement

The exponents can be multiplied when a power is raised to another power.

$$\big(a^b\big)^c = a^{b*c}$$

## Proof

From the definition of powers, write out the factors of $\big(a^b\big)^c$.

$$\big(a^b\big)^c = \underbrace{a^b * a^b * a^b * ...}_\text{c times}$$

Do this again for $a^b$.

$$\big(a^b\big)^c = \underbrace{\overbrace{a * a * ...}^\text{b times} * \overbrace{a * a * ...}^\text{b times} * \overbrace{a * a * ...}^\text{b times} * ...}_\text{c times}$$

Now the factor $a$ is repeated $b * c$ times, thus:

$$\big(a^b\big)^c = a^{b*c}$$

## Proofs building upon this proof

### Change of base formula for logarithms

This proofs shows that every logarithm can be written as a division of two logarithms with a custom base.

### The keep-change-flip rule for fractions

This proof shows the famous keep-change-flip rule when dividing by fractions.

### The logarithm power rule

This proofs shows that a logarithm of a power is the same as the exponent multiplied by that logarithm.