How do you simplify #sqrt(3-sqrt(5+2sqrt(3)))+sqrt(3+sqrt(5+2sqrt(3)))-sqrt((sqrt(3)-1)^2)# ?
1 Answer
Oct 22, 2017
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#