What does it mean when one says that a rational exponent represents both an integer exponent and the root? What if the rational exponent is 3/3?

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In a rational exponent the numerator represents the integer exponent, and the denominator represents the root. For 3/3:

`x^(3/3)=(root(3)(x))^3`

Following the rules of exponents, you can easily simplify this expression to equal x because (3)(1/3)=1.

This is how you end up with rational exponents in the first place. The exponent of the root of a function is 1/root. Therefore, when you multiply an integer exponent to a root exponent you end up with (integer)(1/root)=(integer/root).

To illustrate with another example:

`f(x)=(root(4)(x))^3=(x^(1/4))^3`

Following the rules of exponents that state that a variable to the exponent m raised to the exponent n equals the variable to the power of m times n

`(x^m)^n=x^(mn)`

We get:

`f(x)=(x^(1/4))^3=x^((1/4)(3))=x^(3/4)`

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