Question

What does the denominator of a rational exponent represent

Answer 1

So here's the rule from fractional exponents to radicals:
`x^{(m)/(n)}=\sqrt[n]{x^m}`

Looking at this rule, the denominator represents the root.

Answer 2

The denominator of a rational exponent represents a root.

For example, the exponent
`(1)/(2)` represents a square root. Likewise, an exponent of
`(1)/(3)` represents a cube root.

_____

You can understand this if you recall what a root means, and what the rules of exponents are. A square root multiplied by itself gives the original value. Consider ...

`\displaystyle\left(x^{(1)/(2)}\right)\left(x^{(1)/(2)}\right)=x^{\left((1)/(2)+(1)/(2)\right)}\\\\=x^(1)=x`

For other denominator values, the number of factors that must be multiplied to get x is the number in the denominator—just as you expect for a root.