How do you simplify #sqrt(3-sqrt(5+2sqrt(3)))+sqrt(3+sqrt(5+2sqrt(3)))-sqrt((sqrt(3)-1)^2)# ?

1 Answer
George C.
Oct 22, 2017

#sqrt(3-sqrt(5+2sqrt(3)))+sqrt(3+sqrt(5+2sqrt(3)))-sqrt((sqrt(3)-1)^2) = 2#

Explanation:

Given:

#sqrt(3-sqrt(5+2sqrt(3)))+sqrt(3+sqrt(5+2sqrt(3)))-sqrt((sqrt(3)-1)^2)#

First note that:

#(sqrt(3-sqrt(5+2sqrt(3)))+sqrt(3+sqrt(5+2sqrt(3))))^2#

#=(3-sqrt(5+2sqrt(3)))+2sqrt(9-(5+2sqrt(3)))+(3+sqrt(5+2sqrt(3)))#

#=6+2sqrt(4-2sqrt(3))#

#=6+2sqrt(3-2sqrt(3)+1)#

#=6+2sqrt((sqrt(3)-1)^2)#

#=6+2(sqrt(3)-1)#

#=4+2sqrt(3)#

#=3+2sqrt(3)+1#

#=(sqrt(3)+1)^2#

So:

#sqrt(3-sqrt(5+2sqrt(3)))+sqrt(3+sqrt(5+2sqrt(3)))-sqrt((sqrt(3)-1)^2)#

#=sqrt((sqrt(3)+1)^2)-sqrt((sqrt(3)-1)^2)#

#=(sqrt(3)+1)-(sqrt(3)-1)#

#=2#

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