64^⅔ is a quantity which has a rational exponent, because ⅔ is a rational number.
We can write this is (64^⅓)² or (64²)^⅓ which are 4²=16 or ∛4096=16.
The denominators of the rational exponents mean root, so 1/3 means cube root. The numerators are powers, so 2 means squared.
In this example the exponent and the quantity were both rational—⅔ and 16 are rational numbers.
Now consider 2^⅔. This is the same as (2²)^⅓=4^⅓, which is not rational, but is a radical.
So we have a rational exponent but the quantity is not rational.
Now consider 2^-½, which has a rational exponent -½, but an irrational result which can be written 1/√2.
If we multiply by (√2/√2), which is really multiplying by 1, we get √2/2 which is a better way to write radicals. In general we can write a/√b as a√b/b so that we don’t have a radical in the denominator.
Similarly a/∛b=ab^⅔/b.
Finally, to write a/(b+√c) in a better form, we multiply by (b-√c)/(b-√c): a(b-√c)/(b²-c); and a/(√b+√c)=a(√b-√c)/(b-c). Something a bit more complicated is required to deal with cube root and other radicals. For example, 1/(1+∛2)=⅓(1-∛2+2^⅔).