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Power of a Power Property for Rational Exponents

If a power with base and rational exponent is raised to the power of where is rational, then the result is a power with base and exponent
Multiplication of exponents
The expression can be undefined for some non-positive values of Therefore, this rule will only be defined for positive values of


Power of a Power Property for Rational Exponents
Since and are rational numbers, they can be written as the quotients of two integers. Let and be three integers such that is the quotient of and and is the quotient of and
Using the above definitions, it can be proven that is equal to

Recall that can be defined either as or Using the second definition, the last obtained expression can be rewritten.
The root of a root can be expressed using only one root whose index is the product of the original indices.
Note that since is an integer number, then is also integer. Using the fact that and represent the same expression the power can be moved inside the radical.
As and are integer numbers, the Power of a Power Property can be used.

It has been proven that for rational numbers and