SOLUTION: Express the following in simplest form: {{{root(3,3x^2y^2)/root(3,48x^6y)}}}

Algebra ->  Square-cubic-other-roots -> SOLUTION: Express the following in simplest form: {{{root(3,3x^2y^2)/root(3,48x^6y)}}}       Log On


   

Question 434542: Express the following in simplest form: root%283%2C3x%5E2y%5E2%29%2Froot%283%2C48x%5E6y%29
Answer by Edwin McCravy(20055) About Me  (Show Source):
You can put this solution on YOUR website!
root%283%2C3x%5E2y%5E2%29%2Froot%283%2C48x%5E6y%29

Write as a single cube root:

root%283%2C%283x%5E2y%5E2%29%2F%2848x%5E6y%29%29


Divide top and bottom by 3, which gives 16 on the bottom
Subtract the exponents of x, getting x4 on the bottom.
Subtract the exponents of y, getting y on the top:

root%283%2Cy%2F%2816x%5E4%29%29

rewrite the 16 as 24

root%283%2Cy%2F%282%5E4x%5E4%29%29

The idea is to get the denominator so that all the exponents will
be divisible by 3, the index of the root

To make 2 have an exponent divisible by 3, we need to multiply it
by 22 so it will have an exponent of 6

To make x have an exponent divisible by 3, we need to multiply it
by x2 so it will have an exponent of 6.
So we multiply under the radical by red%28%282x%5E2%29%2F%282x%5E2%29%29


or

root%283%2C%282%5E2x%5E2y%29%2F%282%5E6x%5E6%29%29

Separate into the quotient of two cube roots again:

root%283%2C2%5E2x%5E2y%29%2Froot%283%2C2%5E6x%5E6%29%29

Get rid of the cube root on the bottom by dividing
each exponent by the index 3 of the root:

root%283%2C2%5E2x%5E2y%29%2F%282%5E2x%5E2%29

Changing 22's to 4's

root%283%2C4x%5E2y%29%2F%284x%5E2%29


Edwin