Question

What does the denominator of a rational exponent represent

Answer 1

The denominator of a rational exponent represents a root.

For example, the exponent $\frac{1}{2}$ represents a square root. Likewise, an exponent of $\frac{1}{3}$ represents a cube root.

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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 ...

$\displaystyle\left(x^{\frac{1}{2}}\right)\left(x^{\frac{1}{2}}\right)=x^{\left(\frac{1}{2}+\frac{1}{2}\right)}\\\\=x^{1}=x$

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.

Answer 2

So here's the rule from fractional exponents to radicals: $x^{\frac{m}{n}}=\sqrt[n]{x^m}$

Looking at this rule, the denominator represents the root.