If a power with base a and rational exponentm is raised to the power of n, where n is rational, then the result is a power with base a and exponent m⋅n.
Since m and n are rational numbers, they can be written as the quotients of two integers. Let p,q, and r be three integers such that r=0,m is the quotient of p and r, and n is the quotient of q and r.
m=rpandn=rq
Using the above definitions, it can be proven that (am)n is equal to am⋅n.
Recall that anm can be defined either as nam or (na)m. Using the second definition, the last obtained expression can be rewritten.
(rap)rq=(rrap)q
The root of a root can be expressed using only one root whose index is the product of the original indices.
(rrap)q⇔(r⋅rap)q
Note that since r is an integer number, then r⋅r is also integer. Using the fact that nam and (na)m represent the same expression anm, the power q can be moved inside the radical.