Question 46690
<pre><font size = 5><b>simplify the radical . assume that all variables 
represent positive numbers. 3rd root of 343 x to
the 4th power y to the 5th power
  _______
³<font face = "symbol">Ö</font>343x<sup>4</sup>y<sup>5</sup>

Break everything down to primes

343 = 7·7·7
x<sup>4</sup> = x·x·x·x
y<sup>5</sup> = y·y·y·y·y

So we have this
  _______________________
³<font face = "symbol">Ö</font>7·7·7·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:

  _____________________________
³<font face = "symbol">Ö</font>(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:

  ______________
³<font face = "symbol">Ö</font>7³·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·³<font face = "symbol">Ö</font>x·y·y

or simplifying,
      ___
7xy·³<font face = "symbol">Ö</font>xy² 

Edwin</pre>