(e) \( \frac{x}{x^{\frac{1}{2}}+y^{\frac{1}{2}}}+\frac{x}{x^{\frac{1}{2}}-y^{\frac{1}{2}}} \)

First, let's simplify the given expression: \[ \frac{x}{x^{\frac{1}{2}}+y^{\frac{1}{2}}} + \frac{x}{x^{\frac{1}{2}}-y^{\frac{1}{2}}} \] To combine these fractions, we need a common denominator. The common denominator will be: \[ (x^{\frac{1}{2}} + y^{\frac{1}{2}})(x^{\frac{1}{2}} - y^{\frac{1}{2}}) \] Now we can rewrite both fractions: \[ \frac{x(x^{\frac{1}{2}} - y^{\frac{1}{2}})}{(x^{\frac{1}{2}} + y^{\frac{1}{2}})(x^{\frac{1}{2}} - y^{\frac{1}{2}})} + \frac{x(x^{\frac{1}{2}} + y^{\frac{1}{2}})}{(x^{\frac{1}{2}} - y^{\frac{1}{2}})(x^{\frac{1}{2}} + y^{\frac{1}{2}})} \] Now combine the numerators: \[ \frac{x(x^{\frac{1}{2}} - y^{\frac{1}{2}}) + x(x^{\frac{1}{2}} + y^{\frac{1}{2}})}{(x^{\frac{1}{2}} + y^{\frac{1}{2}})(x^{\frac{1}{2}} - y^{\frac{1}{2}})} \] Simplifying the numerator: \[ = \frac{x(x^{\frac{1}{2}} - y^{\frac{1}{2}} + x^{\frac{1}{2}} + y^{\frac{1}{2}})}{(x^{\frac{1}{2}} + y^{\frac{1}{2}})(x^{\frac{1}{2}} - y^{\frac{1}{2}})} \] This simplifies to: \[ = \frac{x(2x^{\frac{1}{2}})}{(x^{\frac{1}{2}} + y^{\frac{1}{2}})(x^{\frac{1}{2}} - y^{\frac{1}{2}})} \] Thus, the expression now becomes: \[ \frac{2x^{\frac{3}{2}}}{(x^{\frac{1}{2}} + y^{\frac{1}{2}})(x^{\frac{1}{2}} - y^{\frac{1}{2}})} \] Now we can also take into account what happens if we substitute specific values for \( x \) and \( y \); you might discover some interesting behaviors!