Question 154073
<pre><font size = 4 color = "indigo"><b>
{{{(root(3,U)-3root(3,2))/(3root(3,2)-2root(3,U))}}}

To rationalize the denominator when the denominator is the 
difference of cube roots, you must use the factorization of 
the difference of cubes formula.

{{{A^3-B^3=(A-B)(A^2+AB+B^2)}}}

But write the left side on the right and vice-versa, so you
can better see what to do:

{{{(A-B)(A^2+AB+B^2)=A^3-B^3}}}

Let A and B be the terms of the denominator:

{{{A=3root(3,2)}}}, {{{B=2root(3,U)}}}

Substituting into {{{(A-B)(A^2+AB+B^2)=A^3-B^3}}},

{{{(3root(3,2)-2root(3,U))((3root(3,2))^2+3root(3,2)*2root(3,U)+(2root(3,U))^2)=(3root(3,2))^3-(2root(3,U))^3}}}

Simplifying,

{{{3^3(root(3,2))^3-2^3(root(3,U))^3=27*2-8U=54-8U}}}

So as you can see from that, to rationalize the denominator,
you must multiply top and bottom by 

{{{(3root(3,2))^2+3root(3,2)*2root(3,U)+(2root(3,U))^2}}}

which can be simplified a little:

{{{9(root(3,2))^2+6root(3,2U)+4(root(3,U))^2)}}}
{{{9root(3,2^2)+6root(3,2U)+4root(3,U^2)}}}
{{{9root(3,4)+6root(3,2U)+4root(3,U^2)}}}

We have just finished multiplying the bottom by that 
and gotten the rational expression {{{54-8U}}}.

Now we must multiply the numerator, which is

{{{root(3,U)-3root(3,2)}}} 

by it too:

{{{(root(3,U)-3root(3,2))(9root(3,4)+6root(3,2U)+4root(3,U^2))}}}=

{{{9root(3,4U)+6root(3,2U^2)+4root(3,U^3)-27root(3,8)-18root(3,4U)-12root(3,2U^2)}}}

{{{9root(3,4U)+6root(3,2U^2)+4U-27*2-18root(3,4U)-12root(3,2U^2)}}}=

Combine like terms:

{{{-9root(3,4U)-6root(3,2U^2)+4U-54}}}

Now put this over the denominator which we rationalized
first:

{{{(-9root(3,4U)-6root(3,2U^2)+4U-54)/(54-8U)}}}

Edwin</pre></font></b>