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

1 Answer
Jul 23, 2017

x^(1/2)-x^(-1/2) = 2

Explanation:

Note that:

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

Comparing the right hand expression with 3+2sqrt(2), note that it will match if we put a=b=1, so:

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

So 1+sqrt(2) is a square root of 3+2sqrt(2). Since it is positive, it is the principal square root, so:

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

So with x=3+2sqrt(2) we find:

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