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Answer 1 · 2017-08-29T19:07:35

#(13 +3sqrt3 +sqrt5 +2sqrt15 )/11#

The problem is about rationalising the denominator. to do this, multiply with the conjugate in numerator and denominator like this:

#1/(1+sqrt3 -sqrt5) *(1+sqrt3 +sqrt5)/(1+sqrt3 +sqrt5) #

#= (1+sqrt3 +sqrt5)/((1+sqrt3)^2 - (sqrt5)^2)#

#=(1+sqrt3 +sqrt5)/(4+2sqrt3 -5)#

#=(1+sqrt3 +sqrt5)/(2sqrt3 -1)#

Now again multiply with the conjugate

#(1+sqrt3 +sqrt5)/(2sqrt3-1) *(2sqrt3 +1)/(2sqrt3 +1)#

#(2sqrt3 +12 +2sqrt15 +1+sqrt3 +sqrt5)/11#

#(13 +3sqrt3 +sqrt5 +2sqrt15 )/11#