Question 175089
You are describing a process called "Rationalizing The Denominator" which simply means to convert the fraction so that the denominator is a rational number.   The trick is to multiply the fraction by 1 in the form of {{{a/a}}} where {{{a}}} is selected so that after the multiplication the denominator is a rational number.


For example, rationalize the denominator of {{{1/sqrt(2)}}}.  In order for the denominator to be rational, we need to multiply it by {{{sqrt(2)}}}, so that is our value for {{{a}}} in this case.  The process looks like this:


{{{(1/sqrt(2))(sqrt(2)/sqrt(2))=sqrt(2)/2}}}


The process becomes a little more complex when you are faced with a denominator that is in the form {{{a+-b}}} where either {{{a}}} or {{{b}}} (or both, for that matter) is irrational.  In this situation, say if we had {{{2-sqrt(3)}}} as a denominator, there is no simple number that we can use as a multiplier to convert this denominator to a rational number.


What you need to do is to recall the "difference of two squares" factorization, namely:  {{{(a+b)(a-b)=a^2-b^2}}}.  That means you can take something like {{{2-sqrt(3)}}} and multiply by it's conjugate,  {{{2+sqrt(3)}}} to arrive at {{{2^2-sqrt(3)^2=4-3=1}}}


Let's do an example:


Rationalize: {{{2/(3+sqrt(5))}}}


The conjugate of the denominator is {{{3-sqrt(5)}}}, so we have to multiply the original fraction by 1 in the form of {{{(3-sqrt(5))/(3-sqrt(5))}}}, thus:


{{{(2/(3+sqrt(5)))((3-sqrt(5))/(3-sqrt(5)))=(6-2*sqrt(5))/(9-5)=(6-2*sqrt(5))/4=(3-sqrt(5))/2}}}


You can read more about this at:


http://www.math.unt.edu/mathlab/emathlab/How%20to%20Rationalize%20the%20Denominator%20of%20a%20Fraction.htm


http://www.helpalgebra.com/articles/rationalizedenominator.htm


Or just Google "rationalize the denominator" for a bunch more.