1. **State the problem:** Simplify the expression $$\frac{1 + \frac{1}{25}}{6 - 6^{\frac{7}{6}} \cdot \frac{5}{9}}$$. 2. **Simplify the numerator:** $$1 + \frac{1}{25} = \frac{25}{25} + \frac{1}{25} = \frac{26}{25}$$ 3. **Simplify the denominator:** First, calculate the product: $$6^{\frac{7}{6}} \cdot \frac{5}{9}$$ 4. **Rewrite the denominator:** $$6 - 6^{\frac{7}{6}} \cdot \frac{5}{9} = 6 - \frac{5}{9} \cdot 6^{\frac{7}{6}}$$ 5. **Express the entire fraction:** $$\frac{\frac{26}{25}}{6 - \frac{5}{9} \cdot 6^{\frac{7}{6}}} = \frac{26}{25} \div \left(6 - \frac{5}{9} \cdot 6^{\frac{7}{6}}\right)$$ 6. **Rewrite division as multiplication:** $$= \frac{26}{25} \times \frac{1}{6 - \frac{5}{9} \cdot 6^{\frac{7}{6}}}$$ 7. **Final simplified form:** $$\boxed{\frac{26}{25 \left(6 - \frac{5}{9} \cdot 6^{\frac{7}{6}}\right)}}$$ This is the simplified expression. To evaluate numerically, calculate $6^{\frac{7}{6}}$ and substitute.