Definition: A rational exponent is a fractional power: a^(m/n) means the nth root of a, raised to the power m, so the denominator is the root and the numerator is the power.
What does a fractional exponent mean?
An exponent of 1/2 is a square root, and an exponent of 1/3 is a cube root. In general, a1/n is the nth root of a. Then am/n is that root raised to the m.
Why define it this way?
It keeps the exponent rules working. Since (a1/2)2 = a1, the power 1/2 must undo squaring, which is what a square root does.
Example. Evaluate 323/5. The fifth root of 32 is 2, and 2 cubed is 8. So 323/5 = 8. Another: 81-1/4. The fourth root of 81 is 3, and the negative sign means reciprocal, so the value is 1/3.
Which errors appear?
Swapping numerator and denominator, so that the numerator becomes the root. Ignoring the negative sign, which means take a reciprocal and not make the answer negative. Take the root first when you can; it keeps the numbers small, as 82/3 = 22 = 4 and not the cube root of 64 computed with a calculator.