Question

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

Answer 1

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`