problem 9

Notice that :

sqrt (14 + 8sqrt(3) )  =  sqrt ( 14 + sqrt(192)  ]  = sqrt (14 + 2sqrt(48)]  = sqrt (8 + 6 + 2 sqrt(8 * 6) =

sqrt ( 8 + 2sqrt (8 * 6) + 6)  =  8 + 2sqrt(8)sqrt(6) + 6

And the last expression can be factored as :

 [sqrt (8) + sqrt (6)]  * [ sqrt (8) + sqrt(6) ]  =

[sqrt (8) + sqrt6)]^2

So

sqrt (14 + 8sqrt(3) )  =

sqrt [ [sqrt (8) + sqrt6)]^2 ] =

sqrt (8) + sqrt (6)