To simplify radical expressions you are going to need to recognize perfect squares (1, 4, 9, 16, 25, 36, 49, ...), perfect cubes (1, 8, 27, 64, 125, ...), fourth powers (1, 16, 81, 256, 625), 5th powers (1,32, 243,...), etc. An alternative is to factor
the radicand into primes (see http://www.mhhe.com/math/devmath/aleks/wt-bm/student/olc/sl01sec37.htm for a review of this technique). 

Once you have it factored, you can reduce by dividing the index into the exponents.

Find

400 factors into (2 x 2 x 2 x 2 x 5 x 5) or 

The index for cube roots is 3.

Since 3 divides into 4 once with a remainder of 1, one 2 comes out of the radical and one stays under.

Since the exponent 2 is less than the index of 3, 5^2 stays under the radical.

Since 3 goes into 3 once, an x comes out of the radical

Since 3 goes into 7 twice with a remainder of 1, a y^2 comes out and a y stays under the radical.

This gives .

The radical is in simplest form when the exponents under the radical are less than the index of the radical.