Between which two consecutive integers do the following surds lie? \( \begin{array}{ll}\text { 1. } \sqrt{38} & \text { 2. } \sqrt{87} \\ \text { 3. } \sqrt{75} & \text { 4. } \sqrt{43} \\ \text { 5. } \sqrt{150} & \text { 6. } \sqrt{1000} \\ \text { 7. } \sqrt{392} & \text { 8. } \sqrt{700}\end{array} \)

To find the consecutive integers between which each surd lies, we can evaluate the square roots and check against perfect squares. Here’s the breakdown: 1. **For \( \sqrt{38} \)**: The perfect squares around 38 are \( 36 (6^2) \) and \( 49 (7^2) \). Thus, \( \sqrt{38} \) lies between 6 and 7. 2. **For \( \sqrt{87} \)**: The perfect squares are \( 81 (9^2) \) and \( 100 (10^2) \). Therefore, \( \sqrt{87} \) lies between 9 and 10. 3. **For \( \sqrt{75} \)**: The surrounding perfect squares are \( 64 (8^2) \) and \( 81 (9^2) \). Hence, \( \sqrt{75} \) lies between 8 and 9. 4. **For \( \sqrt{43} \)**: The perfect squares are \( 36 (6^2) \) and \( 49 (7^2) \). Consequently, \( \sqrt{43} \) lies between 6 and 7. 5. **For \( \sqrt{150} \)**: The perfect squares are \( 144 (12^2) \) and \( 169 (13^2) \). Thus, \( \sqrt{150} \) lies between 12 and 13. 6. **For \( \sqrt{1000} \)**: The perfect squares are \( 961 (31^2) \) and \( 1024 (32^2) \). So, \( \sqrt{1000} \) lies between 31 and 32. 7. **For \( \sqrt{392} \)**: The perfect squares are \( 361 (19^2) \) and \( 400 (20^2) \). Therefore, \( \sqrt{392} \) lies between 19 and 20. 8. **For \( \sqrt{700} \)**: The perfect squares are \( 676 (26^2) \) and \( 729 (27^2) \). Thus, \( \sqrt{700} \) lies between 26 and 27. In summary: - \( \sqrt{38} \): between 6 and 7 - \( \sqrt{87} \): between 9 and 10 - \( \sqrt{75} \): between 8 and 9 - \( \sqrt{43} \): between 6 and 7 - \( \sqrt{150} \): between 12 and 13 - \( \sqrt{1000} \): between 31 and 32 - \( \sqrt{392} \): between 19 and 20 - \( \sqrt{700} \): between 26 and 27