Multiplying exponents



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

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


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.