SOLUTION: I need help rationalizing.... 9 times the cubic root 2 over/divided by the cubic root 5 minus cubic root 2. Please help! Soory my keyboard will not allow me to type the equati

Algebra ->  Square-cubic-other-roots -> SOLUTION: I need help rationalizing.... 9 times the cubic root 2 over/divided by the cubic root 5 minus cubic root 2. Please help! Soory my keyboard will not allow me to type the equati      Log On


   

Question 203195: I need help rationalizing....
9 times the cubic root 2 over/divided by the cubic root 5 minus cubic root 2.
Please help! Soory my keyboard will not allow me to type the equation.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
%289%2Aroot%283%2C2%29%29%2F%28root%283%2C5%29-root%283%2C2%29%29 Start with the given expression.


Let x=root%283%2C5%29 and y=root%283%2C2%29


%289%2Ay%29%2F%28x-y%29 Replace each root%283%2C5%29 with "x" and each root%283%2C2%29 with "y"


Now this is where you have to get a bit creative. It's very helpful to remember that x%5E3-y%5E3=%28x-y%29%28x%5E2%2Bxy%2By%5E2%29. Why is this important? Because if you cube a cube root, they will both cancel out (leaving the stuff inside as a remainder).

So if we multiply both numerator and denominator by x%5E2%2Bxy%2By%5E2, we get:


%289%2Ay%28x%5E2%2Bxy%2By%5E2%29%29%2F%28%28x-y%29%28x%5E2%2Bxy%2By%5E2%29%29


%289%2Ay%28x%5E2%2Bxy%2By%5E2%29%29%2F%28x%5E3-y%5E3%29 Use the identity given above


Plug in x=root%283%2C5%29 and y=root%283%2C2%29


Cube the cube roots (to eliminate them)


Subtract


Reduce


Combine the roots.


Multiply


3%2Aroot%283%2C2%29%28root%283%2C5%5E2%29%2Broot%283%2C10%29%2Broot%283%2C2%5E2%29%29 Use the identity %28root%28n%2Cx%29%29%5Em=root%28n%2Cx%5Em%29


3%2Aroot%283%2C2%29%28root%283%2C25%29%2Broot%283%2C10%29%2Broot%283%2C4%29%29 Square 5 to get 25 and square 2 to get 4


Distribute


3%2Aroot%283%2C2%2A25%29%2B3%2Aroot%283%2C2%2A10%29%2B3%2Aroot%283%2C2%2A4%29 Combine the roots.


3%2Aroot%283%2C50%29%2B3%2Aroot%283%2C20%29%2B3%2Aroot%283%2C8%29 Multiply


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Answer:


So


This can be verified (as I have verified it myself) with a calculator.