1. **State the problem:** We need to find the area of a square room with side length $2 \frac{1}{5}$ meters. 2. **Formula:** The area $A$ of a square is given by: $$A = s^2$$ where $s$ is the length of one side. 3. **Convert mixed number to improper fraction:** $$2 \frac{1}{5} = \frac{2 \times 5 + 1}{5} = \frac{11}{5}$$ meters. 4. **Calculate the area:** $$A = \left(\frac{11}{5}\right)^2 = \frac{11^2}{5^2} = \frac{121}{25}$$ 5. **Simplify the fraction:** $$\frac{121}{25} = 4 \frac{21}{25} = 4.84$$ meters squared. 6. **Interpretation:** The carpet required to cover the whole room is $4.84$ square meters. **Final answer:** $$\boxed{4.84 \text{ m}^2}$$