Simplifying surds

Express 4/(2+√3+√7) in the form 1+ a√x + b√y where x and y are integers.

Student X

Rationalization is required in this instance, and (a+b)(a-b)= a² -b² shall be used.

 4/(2+√3+√7)= 4/(2+√3+√7) * [(2+√3)-√7]/[(2+√3)-√7]     

                       = 4[(2+√3)-√7]/[ (2+√3)^2 - (√7)^2]         

                       = 4[(2+√3)-√7]/[ 4 + 4√3 + 3 - 7] 

                       = 4[(2+√3)-√7]/4√3 

                       = [(2+√3)-√7]/√3 

                       = [(2+√3)-√7]/√3 * √3/√3 

                       = √3[(2+√3)-√7]/ 3 

                       = 2√3/3 + 1 - √21/3 

                       = √12/3 + 1 - √21/3 

                      = 1 - 1/3*√21 + 1/3*√12  (shown)

Hope this helps. Peace.

Best Regards,

Mr Koh