SOLUTION: root:5:(64x^(8)y^(12)) How do I pull all perfect 5th rotts out?

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Question 242613: root:5:(64x^(8)y^(12))
How do I pull all perfect 5th rotts out?
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
root%285%2C+64x%5E8y%5E12%29

To simplify this we need to find factors that are perfect powers of 5, if any.

For the 64 we look for factors of 64 that are perfect powers of 5, if any. Since 32+=+2%5E5 and 64 = 32*2, we do have a perfect power of 5 factor in 64.

For the variables we can factor out perfect powers of 5 as long as the exponent is greater than or equal to 5.

root%285%2C+64x%5E8y%5E12%29
root%285%2C+32%2A2%2Ax%5E5%2Ax%5E3%2Ay%5E5%2Ay%5E7%29 (There's another power of 5 in y%5E7)
root%285%2C+32%2A2%2Ax%5E5%2Ax%5E3%2Ay%5E5%2Ay%5E5%2Ay%5E2%29
Now we can use the property of radicals, root%28a%2C+p%2Aq%29+=+root%28a%2C+p%29%2Aroot%28a%2C+q%29, to split up all these factors into separate radicals:

Next, for each factor that is a perfect power of 5, we substitute in its value:
2%2Aroot%285%2C+2%29%2Ax%2Aroot%285%2C+x%5E3%29%2Ay%2Ay%2Aroot%285%2C+y%5E2%29
Now we use the Commutative and Associative Properties to rearrange and regroup the factors. We want the non-radical factors grouped together and the radical factors grouped together. (The convention is to put the radical at the end):
%282%2Ax%2Ay%2Ay%29%2A%28root%285%2C+2%29%2Aroot%285%2C+x%5E3%29%2Aroot%285%2C+y%5E2%29%29
which simplifies to:
2xy%5E2%5Croot%285%2C+2x%5E3y%5E2%29