The root of a product is the same as the product of the roots. In other words, a single square root can be broken up across a multiplication sign to be turned into the product of two separate square roots. The same principle applies for cube roots as well as division. A complex fraction inside of a square root sign is equivalent to the same complex fraction with both the numerator and denominator inside a square root.
Radicals are not distributive over addition and subtraction. The square root of a sum or different is not the same as the sum or difference of the square roots.
If one side of an equation contains a radical, both sides of the equation can be squared to eliminate the radical. This will only work if you square the entirety of both sides of the equation. It can get a bit tricky if addition and subtraction are present on the opposite side from the radical. It will not eliminate the radical if there is another term added to or subtracted from the radical.
If you simplify a radical such as rewriting √25 as 5, remember that a square root can be either positive or negative. The square root of x squared can be either x or -x. To account for this possibility, break the problem into two equations, one for the positive case and one for the negative case. Often this will result in two answers to an algebra problem if the variable is inside the radical.
Square roots are the equivalent of an exponent of one-half. Sometimes it can make a problem easier to deal with if square roots are rewritten as the same term to the one-half power. Remember to put the entire term that was in the square root to the one-half power. The same rules for the distributive property apply to exponents as to radicals.