Question 114450
{{{ sqrt( 7 ) - sqrt( 6 )}}} / {{{ sqrt( 7 ) + sqrt( 6 )}}}


To rationalize a denominator that is the sum or difference of two square roots, you need to recall the factorization of the difference of two squares, namely:


{{{(a+b)(a-b)=a^2-b^2}}}


If you multiply the denominator of your fraction by {{{sqrt(7)-sqrt(6)}}}, you will get {{{7-6}}} for your new denominator.  But in order to introduce that factor into the denominator, you have to multiply the entire fraction by 1 in the form of {{{(sqrt(7)-sqrt(6))/(sqrt(7)-sqrt(6))}}}.


Here is the entire expression:  {{{ ((sqrt( 7 ) - sqrt( 6 ))/(sqrt( 7 ) + sqrt( 6 )))*((sqrt(7)-sqrt(6))/(sqrt(7)-sqrt(6)))}}}. 


Now, multiply the binomials and simplify:

{{{ (7-2sqrt(7)sqrt(6)+6)/(7-6)}}} 
{{{1-2sqrt(42)}}}