Answer
$x=3$
Work Step by Step
$x^{1/2}-x^{-1/2}-6x^{-3/2}=0$
$x^{-3/2}(x^{2}-x-6)=0$
$x^{-3/2}(x+2)(x-3)=0$
$x^{-3/2}=0$ or $x+2=0$ or $x-3=0$
$\frac{1}{x^{3/2}}=0$ or $x=-2$ or $x=3$
The left equation has no solution.
The middle solution will not work in the original equation (we can not take the square root of a negative number).
Thus the only solution is $x=3$