What does the denominator of a rational exponent represent

The denominator of a rational exponent represents a root.

For example, the exponent represents a square root. Likewise, an exponent of 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 ...

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.