How do you simplify #\frac { \sqrt(\sqrt { 5} + 2)+\sqrt( \sqrt { 5} - 2} ){\sqrt( \sqrt { 5} + 2} )- \sqrt { 3- 2\sqrt { 2} }#?

1 Answer
George C.
Sep 27, 2017

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

#=sqrt(5)-sqrt(2)#

Explanation:

First note that:

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

So:

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

We find:

#sqrt(sqrt(5)-2)/sqrt(sqrt(5)+2) = (sqrt(sqrt(5)-2))^2/(sqrt(sqrt(5)+2)sqrt(sqrt(5)-2))#

#color(white)(sqrt(sqrt(5)-2)/sqrt(sqrt(5)+2)) = (sqrt(5)-2)/(sqrt((sqrt(5))^2-2^2))#

#color(white)(sqrt(sqrt(5)-2)/sqrt(sqrt(5)+2)) = (sqrt(5)-2)/sqrt(1)#

#color(white)(sqrt(sqrt(5)-2)/sqrt(sqrt(5)+2)) = sqrt(5)-2#

So:

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

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

#=sqrt(5)-sqrt(2)#

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