simplify the radical . assume that all variables 
represent positive numbers. 3rd root of 343 x to
the 4th power y to the 5th power
  _______
³Ö343x4y5

Break everything down to primes

343 = 7·7·7
x4 = x·x·x·x
y5 = y·y·y·y·y

So we have this
  _______________________
³Ö7·7·7·x·x·x·x·y·y·y·y·y

Since it's a cube root (i.e. "third root"),
we use parentheses to group into all possible
groups of three like factors each:

  _____________________________
³Ö(7·7·7)·(x·x·x)·x·(y·y·y)·y·y

Now each of the groups of three can be rewritten
as a cube.  (7·7·7) = 7³, (x·x·x) = x³, and
(y·y·y) = y³,  So now we have:

  ______________
³Ö7³·x³·x·y³·y·y

Now take the cube roots of the cubes under the
radical. That is take the cubes outside in front
of the radical without the cube exponent.  What
did not group stays under the radical.
        _____ 
7·x·y·³Öx·y·y

or simplifying,
      ___
7xy·³Öxy² 

Edwin