1. **Stating the problem:** Calculate the value of the expression $$\frac{1 - 2^{-1} - 3^{-1}}{1 + 2^{-1} + 3^{-1}} \cdot 11$$. 2. **Recall the rules:** - Negative exponents mean reciprocal: $a^{-1} = \frac{1}{a}$. - Perform operations in numerator and denominator separately before multiplying. 3. **Rewrite the expression using reciprocals:** $$\frac{1 - \frac{1}{2} - \frac{1}{3}}{1 + \frac{1}{2} + \frac{1}{3}} \cdot 11$$ 4. **Find common denominators in numerator and denominator:** - Numerator common denominator is 6: $$1 = \frac{6}{6}, \quad \frac{1}{2} = \frac{3}{6}, \quad \frac{1}{3} = \frac{2}{6}$$ - Numerator becomes: $$\frac{6}{6} - \frac{3}{6} - \frac{2}{6} = \frac{6 - 3 - 2}{6} = \frac{1}{6}$$ 5. **Denominator common denominator is also 6:** $$1 = \frac{6}{6}, \quad \frac{1}{2} = \frac{3}{6}, \quad \frac{1}{3} = \frac{2}{6}$$ - Denominator becomes: $$\frac{6}{6} + \frac{3}{6} + \frac{2}{6} = \frac{6 + 3 + 2}{6} = \frac{11}{6}$$ 6. **Rewrite the fraction:** $$\frac{\frac{1}{6}}{\frac{11}{6}} \cdot 11$$ 7. **Divide the fractions:** $$\frac{1}{6} \div \frac{11}{6} = \frac{1}{6} \cdot \frac{6}{11} = \frac{\cancel{1}}{\cancel{6}} \cdot \frac{\cancel{6}}{11} = \frac{1}{11}$$ 8. **Multiply by 11:** $$\frac{1}{11} \cdot 11 = 1$$ **Final answer:** 1