16. Simplify \( \frac{4}{\sqrt{5}+\sqrt{2}}-\frac{3}{\sqrt{5}-\sqrt{2}} \)

To simplify the expression \( \frac{4}{\sqrt{5}+\sqrt{2}}-\frac{3}{\sqrt{5}-\sqrt{2}} \), we can first identify a common denominator. The common denominator of the two fractions is \( (\sqrt{5}+\sqrt{2})(\sqrt{5}-\sqrt{2}) \). We simplify the expression as follows: 1. Multiply the numerators and denominators accordingly: \[ = \frac{4(\sqrt{5}-\sqrt{2})}{(\sqrt{5}+\sqrt{2})(\sqrt{5}-\sqrt{2})} - \frac{3(\sqrt{5}+\sqrt{2})}{(\sqrt{5}+\sqrt{2})(\sqrt{5}-\sqrt{2})} \] 2. The denominator simplifies as follows: \[ (\sqrt{5}+\sqrt{2})(\sqrt{5}-\sqrt{2}) = 5 - 2 = 3 \] 3. Now, we combine the fractions: \[ = \frac{4(\sqrt{5}-\sqrt{2}) - 3(\sqrt{5}+\sqrt{2})}{3} \] 4. Distributing in the numerators gives: \[ = \frac{4\sqrt{5} - 4\sqrt{2} - 3\sqrt{5} - 3\sqrt{2}}{3} \] \[ = \frac{(4\sqrt{5} - 3\sqrt{5}) + (-4\sqrt{2} - 3\sqrt{2})}{3} \] \[ = \frac{\sqrt{5} - 7\sqrt{2}}{3} \] Thus, the simplified expression is: \[ \frac{\sqrt{5} - 7\sqrt{2}}{3} \]