SOLUTION: Rationalize the denominators {{{ sqrt( 7 ) - sqrt( 6 )}}} / {{{ sqrt( 7 ) + sqrt( 6 )}}} Combine {{{ 8sqrt( 12 ) - 3sqrt( 27 ) }}}

Algebra ->  Expressions-with-variables -> SOLUTION: Rationalize the denominators {{{ sqrt( 7 ) - sqrt( 6 )}}} / {{{ sqrt( 7 ) + sqrt( 6 )}}} Combine {{{ 8sqrt( 12 ) - 3sqrt( 27 ) }}}       Log On


   

Question 114450: Rationalize the denominators
+sqrt%28+7+%29+-+sqrt%28+6+%29 / +sqrt%28+7+%29+%2B+sqrt%28+6+%29
Combine
+8sqrt%28+12+%29+-+3sqrt%28+27+%29+

Answer by solver91311(24713) About Me  (Show Source):
You can put this solution on YOUR website!
+sqrt%28+7+%29+-+sqrt%28+6+%29 / +sqrt%28+7+%29+%2B+sqrt%28+6+%29

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:

%28a%2Bb%29%28a-b%29=a%5E2-b%5E2

If you multiply the denominator of your fraction by sqrt%287%29-sqrt%286%29, 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 %28sqrt%287%29-sqrt%286%29%29%2F%28sqrt%287%29-sqrt%286%29%29.

Here is the entire expression: .

Now, multiply the binomials and simplify:
+%287-2sqrt%287%29sqrt%286%29%2B6%29%2F%287-6%29
1-2sqrt%2842%29