Powers of powers

We know how to multiply and divide powers.

We can also do powers of powers.

For example, what is (a²)⁴ ?

Remember that c⁴ = c x c x c x c

     (a²)⁴ = a² x a² x a² x a²

             = a^{2 + 2 + 2 + 2}

             = a⁸

So (a²)⁴ = a^{2 x 4}

This property is general

Hence,

     (a^m)ⁿ = a^mn

And since (aⁿ)^m = a^nm and mn = mn

It follows that

     (a^m)ⁿ = (aⁿ)^m