SOLUTION: {{cube root of 108x^3y^7/2y^3}}

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Question 203101This question is from textbook
: {{cube root of 108x^3y^7/2y^3}} This question is from textbook

Found 2 solutions by jim_thompson5910, jsmallt9:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
root%283%2C%28108x%5E3y%5E7%29%2F%282y%5E3%29%29 Start with the given expression.


root%283%2C%2854x%5E3y%5E7%29%2F%28y%5E3%29%29 Reduce the coefficients.


root%283%2C54x%5E3y%5E%287-3%29%29 Divide the common variable terms by subtracting the exponents.


root%283%2C54x%5E3y%5E4%29 Subtract


root%283%2C27%2A2x%5E3y%5E4%29 Factor 54 into 27*2


root%283%2C27%2A2x%5E3y%5E3%2Ay%29 Factor y%5E4 into y%5E3%2Ay


Break up the root.


3%2Aroot%283%2C2%29%2Aroot%283%2Cx%5E3%29%2Aroot%283%2Cy%5E3%29%2Aroot%283%2Cy%29 Take the cube root of 27 to get 3


3%2Aroot%283%2C2%29%2Ax%2Aroot%283%2Cy%5E3%29%2Aroot%283%2Cy%29 Take the cube root of x%5E3 to get "x"


3%2Aroot%283%2C2%29%2Ax%2Ay%2Aroot%283%2Cy%29 Take the cube root of y%5E3 to get "y"


3xy%2Aroot%283%2C2y%29 Rearrange the terms and combine the roots.


So root%283%2C%28108x%5E3y%5E7%29%2F%282y%5E3%29%29=3xy%2Aroot%283%2C2y%29 where y%3C%3E0

Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
root%283%2C+%28108x%5E3%2Ay%5E7%29%2F%282y%5E3%29%29
Let's start by reducing the fraction inside:

Now we look for perfect cube factors:

Now we can use one of the properties of roots (which apply for all roots: square, cube, 4th, 5th, etc.): sqrt%28x%2Ay%29+=+sqrt%28x%29%2Asqrt%28y%29. This allows us to separate factors in a root into separate roots:

Now we can simplify the cube roots of the perfect cubes:

Using the Commutative Property of multiplication to rearrange the order:

And finally, using the same property of roots as earlier (except in reverse), to combine the cube roots:
3xy%2Aroot%283%2C+2%29%2Aroot%283%2C+y%29+=+3xy%2Aroot%283%2C+2y%29