Explain how to estimate 17. \( \quad \sqrt{\mathbf{1 7 5}} \)

To estimate \( \sqrt{175} \), we can use a few simple steps: 1. **Identify Perfect Squares**: First, we find the perfect squares that are close to 175. The perfect squares around 175 are: - \( 12^2 = 144 \) - \( 13^2 = 169 \) - \( 14^2 = 196 \) Since \( 175 \) is between \( 13^2 \) and \( 14^2 \), we know that: \[ 13 < \sqrt{175} < 14 \] 2. **Refine the Estimate**: To get a better estimate, we can check the midpoint between 13 and 14, which is 13.5: \[ 13.5^2 = 182.25 \] Since \( 182.25 > 175 \), we know that: \[ \sqrt{175} < 13.5 \] Now we can check a value between 13 and 13.5, such as 13.3: \[ 13.3^2 = 176.89 \] Since \( 176.89 > 175 \), we know that: \[ \sqrt{175} < 13.3 \] Next, we can try 13.2: \[ 13.2^2 = 174.24 \] Since \( 174.24 < 175 \), we know that: \[ 13.2 < \sqrt{175} \] 3. **Final Estimate**: Now we have: \[ 13.2 < \sqrt{175} < 13.3 \] Therefore, we can estimate that: \[ \sqrt{175} \approx 13.3 \] In conclusion, the estimated value of \( \sqrt{175} \) is approximately \( 13.3 \).