I teach 3 steps/rules for simplifying radicals:

(1) There can be no fractions in the radicand.

Use the "rule" `sqrt(a/b)=sqrt(a)/sqrt(b)` (True for any index -- thus `root(3)(a/b)=(root(3)(a))/(root(3)(b))` etc...)

(2) There can be no radicals in the denominator.

Use conjugation to clear the radical from the denominator.

`7/sqrt(2)=7/sqrt(2)*sqrt(2)/sqrt(2)=(7sqrt(2))/2`

This is somewhat complicated for other indices. We multiply both numerator and denominator by a radical that creates a perfect nth power in the radicand of the denominator where n is the index.

`7/root(3)(3)=7/root(3)(3)*root(3)(9)/root(3)(9)=(7root(3)(9))/3`

(3) There can be no perfect nth powers in the radicand where n is the index.

`sqrt(18x^4y^5)=sqrt(9*2*x^4*y^4*y)=sqrt(9x^4y^4)sqrt(2y)=3x^2y^2sqrt(2y)`

`root(3)(54)=root(3)(27*2)=root(3)(27)root(3)(2)=3root(3)(2)`