Power of a Power Property

To find a power of a power, multiply the exponents.  This is an extension of the product of powers property . Suppose you have a number raised to a power, and you multiply the whole expression by itself over and over. This is the same as raising the expression to a power:

${\left(5^3\right)}^4=\left(5^3\right)\left(5^3\right)\left(5^3\right)\left(5^3\right)$

But the product of powers property tells us that

${\left(5^3\right)}^4=\left(5^3\right)\left(5^3\right)\left(5^3\right)\left(5^3\right)=5^{3+3+3+3}=5^{4\left(3\right)}=5^{12}$

So it is enough to just multiply the powers!

In general,

${\left(a^b\right)}^c=a^{bc}$ .