Simplify the expression. \[ \frac{9}{1-\sqrt{19}} \]

To simplify the expression: \[ \frac{9}{1 - \sqrt{19}} \] we can rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(1 - \sqrt{19}\) is \(1 + \sqrt{19}\). **Step-by-Step Simplification:** 1. **Multiply by the Conjugate:** \[ \frac{9}{1 - \sqrt{19}} \times \frac{1 + \sqrt{19}}{1 + \sqrt{19}} = \frac{9(1 + \sqrt{19})}{(1 - \sqrt{19})(1 + \sqrt{19})} \] 2. **Simplify the Denominator:** \[ (1 - \sqrt{19})(1 + \sqrt{19}) = 1^2 - (\sqrt{19})^2 = 1 - 19 = -18 \] 3. **Combine the Results:** \[ \frac{9(1 + \sqrt{19})}{-18} = \frac{9}{-18}(1 + \sqrt{19}) = -\frac{1}{2}(1 + \sqrt{19}) \] 4. **Final Simplified Form:** \[ -\frac{1 + \sqrt{19}}{2} \] **Therefore, the simplified expression is:** \[ -\frac{1 + \sqrt{19}}{2} \] **Alternatively, it can be written as:** \[ \frac{ -1 - \sqrt{19} }{2 } \] Either form is acceptable and correct. **Final Answer:** \[ -\frac{\,1+\sqrt{19}\,}{\,2\,} \]