1. **State the problem:** Solve for $ab$ given the equations: $$(a - 6)(a + 6) = 49$$ $$a^2 + 6^2 = 337$$ $$\sqrt{a} + \sqrt{6} = 7$$ and find the value of $$\frac{ab}{\sqrt{a} - \sqrt{6}}$$. 2. **Use the difference of squares formula:** $$(a - 6)(a + 6) = a^2 - 6^2 = 49$$ 3. **Substitute $6^2 = 36$ and simplify:** $$a^2 - 36 = 49$$ 4. **Add 36 to both sides:** $$a^2 = 49 + 36 = 85$$ 5. **Check the second equation:** $$a^2 + 6^2 = 337$$ Substitute $6^2 = 36$: $$a^2 + 36 = 337$$ 6. **Subtract 36 from both sides:** $$a^2 = 337 - 36 = 301$$ 7. **Notice the contradiction:** From step 4, $a^2 = 85$, but from step 6, $a^2 = 301$. This is inconsistent, so we must re-examine the problem. 8. **Re-examine the problem:** The first equation is $(a - 6)(a + 6) = 49$, which simplifies to $a^2 - 36 = 49$, so $a^2 = 85$. The second equation is $a^2 + 6^2 = 337$, which is $a^2 + 36 = 337$, so $a^2 = 301$. Since these two values for $a^2$ differ, the problem likely involves different variables or a misinterpretation. 9. **Assuming the second equation is about $a^2 + b^2 = 337$ instead of $a^2 + 6^2$, and $b=6$, then:** $$a^2 + b^2 = 337$$ $$a^2 + 6^2 = 337$$ $$a^2 + 36 = 337$$ $$a^2 = 301$$ 10. **From the first equation, $(a - 6)(a + 6) = 49$, which is $a^2 - 36 = 49$, so:** $$a^2 = 85$$ This contradicts the previous $a^2 = 301$. 11. **Therefore, the problem likely involves different variables $a$ and $b$ with $b=6$. Let's denote $b=6$ and $a$ unknown. Then:** From the first equation: $$(a - b)(a + b) = 49$$ $$a^2 - b^2 = 49$$ From the second equation: $$a^2 + b^2 = 337$$ 12. **Add the two equations:** $$(a^2 - b^2) + (a^2 + b^2) = 49 + 337$$ $$2a^2 = 386$$ $$a^2 = 193$$ 13. **Subtract the first equation from the second:** $$(a^2 + b^2) - (a^2 - b^2) = 337 - 49$$ $$2b^2 = 288$$ $$b^2 = 144$$ $$b = 12$$ 14. **Calculate $a$:** $$a = \sqrt{193}$$ 15. **Given $\sqrt{a} + \sqrt{6} = 7$, solve for $\sqrt{a}$:** $$\sqrt{a} = 7 - \sqrt{6}$$ 16. **Square both sides:** $$a = (7 - \sqrt{6})^2 = 49 - 2 \times 7 \times \sqrt{6} + 6 = 55 - 14\sqrt{6}$$ 17. **Equate this to $a = \sqrt{193}$ from step 14, which is inconsistent, so the problem likely means $\sqrt{a}$ is a number, not $a$ itself. Let's denote $x = \sqrt{a}$, then:** $$x + \sqrt{6} = 7$$ $$x = 7 - \sqrt{6}$$ 18. **Calculate $ab$ using the identity:** $$(x + y)(x - y) = x^2 - y^2$$ Here, $x = \sqrt{a}$ and $y = \sqrt{6}$. 19. **Calculate $\sqrt{a} - \sqrt{6}$:** $$\sqrt{a} - \sqrt{6} = x - y = (7 - \sqrt{6}) - \sqrt{6} = 7 - 2\sqrt{6}$$ 20. **Calculate $ab$ using:** $$ab = (\sqrt{a})^2 \times (\sqrt{b})^2 = a \times b$$ Given $b = 6$, and $a = x^2 = (7 - \sqrt{6})^2 = 55 - 14\sqrt{6}$. So: $$ab = a \times b = (55 - 14\sqrt{6}) \times 6 = 330 - 84\sqrt{6}$$ 21. **Calculate the expression:** $$\frac{ab}{\sqrt{a} - \sqrt{6}} = \frac{330 - 84\sqrt{6}}{7 - 2\sqrt{6}}$$ 22. **Rationalize the denominator:** Multiply numerator and denominator by the conjugate of the denominator: $$\frac{330 - 84\sqrt{6}}{7 - 2\sqrt{6}} \times \frac{7 + 2\sqrt{6}}{7 + 2\sqrt{6}} = \frac{(330 - 84\sqrt{6})(7 + 2\sqrt{6})}{7^2 - (2\sqrt{6})^2}$$ 23. **Calculate denominator:** $$49 - 4 \times 6 = 49 - 24 = 25$$ 24. **Calculate numerator:** $$330 \times 7 + 330 \times 2\sqrt{6} - 84\sqrt{6} \times 7 - 84\sqrt{6} \times 2\sqrt{6}$$ $$= 2310 + 660\sqrt{6} - 588\sqrt{6} - 168 \times 6$$ $$= 2310 + (660 - 588)\sqrt{6} - 1008$$ $$= 2310 + 72\sqrt{6} - 1008 = 1302 + 72\sqrt{6}$$ 25. **Final expression:** $$\frac{1302 + 72\sqrt{6}}{25} = \frac{1302}{25} + \frac{72}{25}\sqrt{6}$$ **Answer:** $$\boxed{\frac{1302}{25} + \frac{72}{25}\sqrt{6}}$$