Only one kind of root is called a "SQUARE" root. Sometimes students
get the wrong impression that all roots are called "SQUARE" roots, but
this is false.
simplify by factoring 5 square root 96x^12y^15
Did you mean this 
?
If so that is NOT a "SQUARE" root but a FIFTH root.
If it were as you stated it would be 
which is an entirely different problem.
If you meant the FIFTH root, not the SQUARE root, then you change
to 
because 10 is the largest exponent
divisible by 5. We also change 
to 
or 
.
We do not need to change 
because 15 is already divisible
by 5, the index of the 5th root.
= 
=
=
Now we divide each exponent that is divisible by 5 and bring it out
from under the radical with the exponent divided by the index 5:
and you can drop the 1 exponent for x
If you really meant "SQUARE ROOT", like this:
Then the index is understood to be 2. I'll put the index:
then you do not need to change because 12 is already
divisible by 2. We change to 
or 
, so
that the exponent 4 is divisible by 2.
We change to 
.
Now we divide each exponent by 2 that is divisible by 2 and bring it out
from under the radical with the exponent divided by the index 2:
=
and we drop the 2 index because it's understood when the root is a
SQUARE root and not some other root such as the FIFTH root.
=
Edwin
