1. The problem is to simplify the expression $\frac{\sqrt{5-7}}{\sqrt{5}+\sqrt{7}}$. 2. First, simplify inside the square root in the numerator: $5 - 7 = -2$. 3. So the expression becomes $\frac{\sqrt{-2}}{\sqrt{5}+\sqrt{7}}$. 4. Since $\sqrt{-2} = i\sqrt{2}$ where $i$ is the imaginary unit, the expression is now $\frac{i\sqrt{2}}{\sqrt{5}+\sqrt{7}}$. 5. To simplify further, rationalize the denominator by multiplying numerator and denominator by the conjugate $\sqrt{5}-\sqrt{7}$: $$\frac{i\sqrt{2}}{\sqrt{5}+\sqrt{7}} \times \frac{\sqrt{5}-\sqrt{7}}{\sqrt{5}-\sqrt{7}} = \frac{i\sqrt{2}(\sqrt{5}-\sqrt{7})}{(\sqrt{5}+\sqrt{7})(\sqrt{5}-\sqrt{7})}$$ 6. The denominator uses difference of squares formula: $$(\sqrt{5})^2 - (\sqrt{7})^2 = 5 - 7 = -2$$ 7. Substitute back: $$\frac{i\sqrt{2}(\sqrt{5}-\sqrt{7})}{-2} = \frac{i\sqrt{2}(\sqrt{5}-\sqrt{7})}{-2}$$ 8. Simplify the negative in denominator: $$-\frac{i\sqrt{2}(\sqrt{5}-\sqrt{7})}{2} = \frac{-i\sqrt{2}(\sqrt{5}-\sqrt{7})}{2}$$ Final answer: $$\boxed{\frac{-i\sqrt{2}(\sqrt{5}-\sqrt{7})}{2}}$$