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

##### 1 Answer

Jul 23, 2017

#### Explanation:

Note that:

#(a+bsqrt(2))^2 = (a^2+2b^2)+2ab sqrt(2)#

Comparing the right hand expression with

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

So

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

So with

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

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

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

#color(white)(x^(1/2)-x^(-1/2)) = (1+color(red)(cancel(color(black)(sqrt(2)))))+(1-color(red)(cancel(color(black)(sqrt(2)))))#

#color(white)(x^(1/2)-x^(-1/2)) = 2#