Because sometimes this makes UNLIKE terms become LIKE terms,
and whereas UNLIKE terms cannot be combined, LIKE terms can.

Example:
 __     __
∛24 + ∛81

is an expression with unlike terms and they cannot be combined 
if we left them that way. However if we break 24 down as 8*3 
and 81 down as 27*3, like this:

 ___     ____
∛8*3 + ∛27*3

then since the cube root of 8 is 2 and the cube root of 27 is 3,
we can just put their cube roots in front of the radicals
and erase them under the radicals:
  __     _
2∛3 + 3∛3

Now we have two LIKE terms and we can now combine them and
end up with:
  __ 
5∛3

Edwin