Simplify \( \frac{\sqrt{75}-\sqrt{50}}{(\sqrt{10}-\sqrt{5})(\sqrt{10}+\sqrt{5})} \)

To simplify the expression \( \frac{\sqrt{75}-\sqrt{50}}{(\sqrt{10}-\sqrt{5})(\sqrt{10}+\sqrt{5})} \), we start by simplifying the denominator. The denominator \( (\sqrt{10}-\sqrt{5})(\sqrt{10}+\sqrt{5}) \) is a difference of squares, which simplifies to: \[ (\sqrt{10})^2 - (\sqrt{5})^2 = 10 - 5 = 5. \] Now, we can rewrite the original expression as: \[ \frac{\sqrt{75} - \sqrt{50}}{5}. \] Next, we simplify the numerator \( \sqrt{75} - \sqrt{50} \). We can express \( \sqrt{75} \) and \( \sqrt{50} \) in terms of simpler square roots: \[ \sqrt{75} = \sqrt{25 \cdot 3} = 5\sqrt{3}, \] \[ \sqrt{50} = \sqrt{25 \cdot 2} = 5\sqrt{2}. \] Thus, the numerator becomes: \[ 5\sqrt{3} - 5\sqrt{2} = 5(\sqrt{3} - \sqrt{2}). \] Now we can substitute back into the expression: \[ \frac{5(\sqrt{3}-\sqrt{2})}{5}. \] The \( 5 \) in the numerator and denominator cancels out: \[ \sqrt{3} - \sqrt{2}. \] Therefore, the simplified form of the expression is: \[ \boxed{\sqrt{3} - \sqrt{2}}. \]