1. **State the problem:** We need to evaluate the square root of 49 and add the two mixed fractions $2 \frac{5}{8}$ and $2 \frac{3}{8}$.\n\n2. **Evaluate the square root:** The square root of 49 is the number which when multiplied by itself gives 49.\n$$\sqrt{49} = 7$$\n\n3. **Add the mixed fractions:** First, convert the mixed fractions to improper fractions.\n$$2 \frac{5}{8} = \frac{2 \times 8 + 5}{8} = \frac{16 + 5}{8} = \frac{21}{8}$$\n$$2 \frac{3}{8} = \frac{2 \times 8 + 3}{8} = \frac{16 + 3}{8} = \frac{19}{8}$$\n\n4. **Add the improper fractions:** Since denominators are the same, add the numerators.\n$$\frac{21}{8} + \frac{19}{8} = \frac{21 + 19}{8} = \frac{40}{8}$$\n\n5. **Simplify the fraction:**\n$$\frac{40}{8} = \cancel{\frac{40}{8}}^{\frac{40 \div 8}{8 \div 8}} = 5$$\n\n6. **Final answers:**\n- Square root of 49 is $7$.\n- Sum of $2 \frac{5}{8}$ and $2 \frac{3}{8}$ is $5$.