Do you like flowers, alwaysalillost? becuase thinking of flowers is the easiest way to do this. It may seem cheezy to you, but to this day, I use this rule to double check myself that I am changing from exponents to roots and vice versa correctly.
It is known as Flower Power!
Basically, we equate a fractional exponent to a flower. The top number represents the flower (which rhymes with power) and the bottom number represents the root, and the division sign represents the ground. So now we know that whenever we have a fractional exponent, the top number is the power of the base and the bottom number is the root of the base.
so sqrt(x) = x^(1/2) because the power of x is 1, and the root we are using is the 2nd root.
fourthroot(2) = 2^(1/4), since the power of 2 is 1 and the root is the fourth
the cuberoot(x^2*y^3) = cuberoot(x^2)*cuberoot(y^3) by the law of surds.
now cuberoot(x^2) = x^(2/3) since the (flower) power of x is 2 so it goes on top, and the root is the third root, so it goes on the bottom.
and cuberoot(y^3) = y^(3/3) since the power of y is 3 and the root of y is 3, so we get y^1 or just y
therefore, cuberoot(x^2*y^3) = cuberoot(x^2)*cuberoot(y^3) = (x^(2/3))*y
Note: we can take the root or the power first.
that is, for example, if we have x^(2/3) we can write it as cuberoot(x^2) or [cuberoot(x)]^2 depending on which woud be easier to calculate. if you want an example of when one would be easier than the other, ask