1. **State the problem:** Approximate the square roots of given numbers using nearby perfect squares and interpolation. 2. **Recall the perfect squares:** - $3^2 = 9$ - $4^2 = 16$ - $5^2 = 25$ 3. **Approximate each square root by finding where the number lies between perfect squares:** **a. $\sqrt{11}$:** - $11$ is between $9$ and $16$ - Distance from $9$ to $11$ is $2$, total interval is $7$ - Approximate: $3 + \frac{2}{7} = 3.2857$ **b. $\sqrt{14}$:** - $14$ is between $9$ and $16$ - Distance from $9$ to $14$ is $5$, total interval is $7$ - Approximate: $3 + \frac{5}{7} = 3.7143$ **c. $\sqrt{18}$:** - $18$ is between $16$ and $25$ - Distance from $16$ to $18$ is $2$, total interval is $9$ - Approximate: $4 + \frac{2}{9} = 4.2222$ **d. $\sqrt{24}$:** - $24$ is between $16$ and $25$ - Distance from $16$ to $24$ is $8$, total interval is $9$ - Approximate: $4 + \frac{8}{9} = 4.8889$ 4. **For larger numbers, use nearby perfect squares:** - $15^2 = 225$ - $10^2 = 100$ - $9^2 = 81$ - $4^2 = 16$ - $5^2 = 25$ **a. $\sqrt{222}$:** - Between $15^2=225$ and $14^2=196$ - Distance from $196$ to $222$ is $26$, total interval $29$ - Approximate: $14 + \frac{26}{29} = 14.8966$ **b. $\sqrt{104}$:** - Between $10^2=100$ and $11^2=121$ - Distance from $100$ to $104$ is $4$, total interval $21$ - Approximate: $10 + \frac{4}{21} = 10.1905$ **c. $\sqrt{85}$:** - Between $9^2=81$ and $10^2=100$ - Distance from $81$ to $85$ is $4$, total interval $19$ - Approximate: $9 + \frac{4}{19} = 9.2105$ **d. $\sqrt{17.2}$:** - Between $4^2=16$ and $5^2=25$ - Distance from $16$ to $17.2$ is $1.2$, total interval $9$ - Approximate: $4 + \frac{1.2}{9} = 4.1333$ **e. $\sqrt{157}$:** - Between $12^2=144$ and $13^2=169$ - Distance from $144$ to $157$ is $13$, total interval $25$ - Approximate: $12 + \frac{13}{25} = 12.52$ **f. $\sqrt{26}$:** - Between $5^2=25$ and $6^2=36$ - Distance from $25$ to $26$ is $1$, total interval $11$ - Approximate: $5 + \frac{1}{11} = 5.0909$ **Final answers:** - $\sqrt{11} \approx 3.29$ - $\sqrt{14} \approx 3.71$ - $\sqrt{18} \approx 4.22$ - $\sqrt{24} \approx 4.89$ - $\sqrt{222} \approx 14.90$ - $\sqrt{104} \approx 10.19$ - $\sqrt{85} \approx 9.21$ - $\sqrt{17.2} \approx 4.13$ - $\sqrt{157} \approx 12.52$ - $\sqrt{26} \approx 5.09$