If #x=3+2sqrt2# Then what is # x^(1/2)−x^(−1/2)#?

1 Answer
Zor Shekhtman
Feb 10, 2015

The answer is #x=2#.

Notice that
#x=3+2sqrt(2)=2+2sqrt(2)+1=(sqrt(2)+1)^2#

Therefore, #sqrt(x)=sqrt(2)+1#.
#x^(1/2)-x^(-1/2)=sqrt(x)-1/sqrt(x)=sqrt(2)+1-1/(sqrt(2)+1)#

The last expression can be transformed:
#sqrt(2)+1-1/(sqrt(2)+1) =[(sqrt(2)+1)^2-1]/(sqrt(2)+1)#

Opening parenthesis,
#(2+2sqrt(2)+1-1)/(sqrt(2)+1)=2(sqrt(2)+1)/(sqrt(2)+1=2#

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