Rational exponents

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A rational exponent indicates that the expression is raised to two different powers, one an integer and the other the reciprocal of an integer. These powers can be applied in either of two orders; i.e., either power can be applied first, with the other applied after that.

For example x^(4/3) can be written as either

(x^4)^(1/3)

or as

(x^(1/3)) ^ 4.

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Now if the expression inside has multiple factors, some with exponents of their own, we can get something like

(x^2 y^3 / z^5) ^ (4/3)

I expect this is the sort of expression you're referring to.

We can express this as

(x^2)^(4/3) * y^3 ^ (4/3) / (z^5)^(4/3),

using the laws of exponents. Applying the laws of exponents to this we can get

x^(8/3) * y^(12/3) / z^(20/3).

Now y^(12/3) is just y^4.

x^(8/3) can be written as x^(6/3 + 2/3) = x^(2 + 2/3) = x^2 * x^(2/3).

z^(20/3) can be written as z^(18/3 + 2/3) = z^(6 + 2/3) = z^6 * z^(2/3).

So our expression can be written as

x^2 x^(2/3) y^4 / (z^6 z^(2/3) )

= x^2 y^2 / z^2 * x^(2/3) / y^(2/3)

= x^2 y^2 / z^2 * (x/y)^(2/3).

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