1. **State the problem:** Simplify the expression $$\frac{2\frac{1}{2}}{1-\left(\frac{1}{2}\right)^2}$$. 2. **Convert mixed number to improper fraction:** $$2\frac{1}{2} = 2 + \frac{1}{2} = \frac{4}{2} + \frac{1}{2} = \frac{5}{2}$$ 3. **Rewrite the expression:** $$\frac{\frac{5}{2}}{1 - \left(\frac{1}{2}\right)^2}$$ 4. **Calculate the denominator:** $$\left(\frac{1}{2}\right)^2 = \frac{1}{4}$$ $$1 - \frac{1}{4} = \frac{4}{4} - \frac{1}{4} = \frac{3}{4}$$ 5. **Rewrite the expression with simplified denominator:** $$\frac{\frac{5}{2}}{\frac{3}{4}}$$ 6. **Divide by a fraction by multiplying by its reciprocal:** $$\frac{5}{2} \times \frac{4}{3}$$ 7. **Multiply numerators and denominators:** $$\frac{5 \times 4}{2 \times 3} = \frac{20}{6}$$ 8. **Simplify the fraction by canceling common factors:** $$\frac{\cancel{20}^{10} \times 2}{\cancel{6}^{3} \times 2} = \frac{10}{3}$$ 9. **Final answer:** $$\frac{10}{3}$$