1. **State the problem:** Simplify the expression $$\frac{2+\sqrt{5}}{5-\sqrt{7}} \cdot \frac{5-\sqrt{7}}{5-\sqrt{7}}$$ and verify if it equals $$\frac{\sqrt{5}+7}{8}$$. 2. **Understand the operation:** Multiplying by $$\frac{5-\sqrt{7}}{5-\sqrt{7}}$$ is multiplying by 1, so it should not change the value but rationalize the denominator. 3. **Multiply the numerators:** $$ (2+\sqrt{5})(5-\sqrt{7}) = 2 \cdot 5 - 2 \cdot \sqrt{7} + \sqrt{5} \cdot 5 - \sqrt{5} \cdot \sqrt{7} = 10 - 2\sqrt{7} + 5\sqrt{5} - \sqrt{35} $$ 4. **Multiply the denominators:** $$ (5-\sqrt{7})(5-\sqrt{7}) = (5)^2 - 2 \cdot 5 \cdot \sqrt{7} + (\sqrt{7})^2 = 25 - 10\sqrt{7} + 7 = 32 - 10\sqrt{7} $$ 5. **Rewrite the expression:** $$ \frac{10 - 2\sqrt{7} + 5\sqrt{5} - \sqrt{35}}{32 - 10\sqrt{7}} $$ 6. **Check if this equals:** $$ \frac{\sqrt{5} + 7}{8} $$ 7. **Cross-multiply to verify equality:** $$ (10 - 2\sqrt{7} + 5\sqrt{5} - \sqrt{35}) \cdot 8 \stackrel{?}{=} (\sqrt{5} + 7)(32 - 10\sqrt{7}) $$ 8. **Left side:** $$ 8 \times 10 = 80 $$ $$ 8 \times (-2\sqrt{7}) = -16\sqrt{7} $$ $$ 8 \times 5\sqrt{5} = 40\sqrt{5} $$ $$ 8 \times (-\sqrt{35}) = -8\sqrt{35} $$ Sum left side: $$ 80 - 16\sqrt{7} + 40\sqrt{5} - 8\sqrt{35} $$ 9. **Right side:** Expand: $$ (\sqrt{5} + 7)(32 - 10\sqrt{7}) = \sqrt{5} \times 32 - \sqrt{5} \times 10\sqrt{7} + 7 \times 32 - 7 \times 10\sqrt{7} $$ Calculate: $$ 32\sqrt{5} - 10\sqrt{35} + 224 - 70\sqrt{7} $$ 10. **Compare both sides:** Left: $$80 - 16\sqrt{7} + 40\sqrt{5} - 8\sqrt{35}$$ Right: $$224 - 70\sqrt{7} + 32\sqrt{5} - 10\sqrt{35}$$ They are not equal, so the original equality does not hold. **Final answer:** The expression $$\frac{2+\sqrt{5}}{5-\sqrt{7}} \cdot \frac{5-\sqrt{7}}{5-\sqrt{7}}$$ simplifies to $$\frac{10 - 2\sqrt{7} + 5\sqrt{5} - \sqrt{35}}{32 - 10\sqrt{7}}$$ and does NOT equal $$\frac{\sqrt{5} + 7}{8}$$.