Simplify: (sqrt(2) + i) / (sqrt(2) + sqrt(5))

To simplify the expression, we need to rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator.

Step 1: Multiply the numerator and denominator by the conjugate of the denominator. The denominator is $\sqrt{2}+\sqrt{5}$, so its conjugate is $\sqrt{2}-\sqrt{5}$. $\frac{\sqrt{2}+1}{\sqrt{2}+\sqrt{5}} \times \frac{\sqrt{2}-\sqrt{5}}{\sqrt{2}-\sqrt{5}}$

Step 2: Expand the numerator. $(\sqrt{2}+1)(\sqrt{2}-\sqrt{5}) = \sqrt{2} \cdot \sqrt{2} - \sqrt{2} \cdot \sqrt{5} + 1 \cdot \sqrt{2} - 1 \cdot \sqrt{5}$ $= 2 - \sqrt{10} + \sqrt{2} - \sqrt{5}$

Step 3: Expand the denominator using the difference of squares formula $(a+b)(a-b) = a^2 - b^2$. $(\sqrt{2}+\sqrt{5})(\sqrt{2}-\sqrt{5}) = (\sqrt{2})^2 - (\sqrt{5})^2$ $= 2 - 5$ $= -3$

Step 4: Combine the expanded numerator and denominator. $\frac{2 - \sqrt{10} + \sqrt{2} - \sqrt{5}}{-3}$

Step 5: Rewrite the expression by moving the negative sign to the numerator or in front of the fraction. $\frac{-(2 - \sqrt{10} + \sqrt{2} - \sqrt{5})}{3} = \frac{-2 + \sqrt{10} - \sqrt{2} + \sqrt{5}}{3}$ Rearranging the terms in the numerator: \frac{\sqrt{10 + \sqrt{5} - \sqrt{2} - 2}{3}}