1. Between which two integers do the following surds lie? \( \begin{array}{ll}\text { a) } \sqrt{56} & \text { b) }-\sqrt{12} \\ \text { c) }-\sqrt{78} & \text { d) } \sqrt{15} \\ \text { e) }-\sqrt{43} & \text { f) }-\sqrt{29} \\ \text { g) } \sqrt{99} & \text { h) } \sqrt{8}\end{array} \)

Let's determine between which two consecutive integers each surd lies by evaluating their approximate values. ### a) \( \sqrt{56} \) - **Calculation**: \(7^2 = 49\) and \(8^2 = 64\). - **Conclusion**: \(7 < \sqrt{56} < 8\). ### b) \( -\sqrt{12} \) - **Calculation**: \( \sqrt{12} \approx 3.464 \). - **Conclusion**: \(-4 < -\sqrt{12} < -3\). ### c) \( -\sqrt{78} \) - **Calculation**: \( \sqrt{78} \approx 8.832 \). - **Conclusion**: \(-9 < -\sqrt{78} < -8\). ### d) \( \sqrt{15} \) - **Calculation**: \(3^2 = 9\) and \(4^2 = 16\). - **Conclusion**: \(3 < \sqrt{15} < 4\). ### e) \( -\sqrt{43} \) - **Calculation**: \( \sqrt{43} \approx 6.557 \). - **Conclusion**: \(-7 < -\sqrt{43} < -6\). ### f) \( -\sqrt{29} \) - **Calculation**: \( \sqrt{29} \approx 5.385 \). - **Conclusion**: \(-6 < -\sqrt{29} < -5\). ### g) \( \sqrt{99} \) - **Calculation**: \(9^2 = 81\) and \(10^2 = 100\). - **Conclusion**: \(9 < \sqrt{99} < 10\). ### h) \( \sqrt{8} \) - **Calculation**: \(2^2 = 4\) and \(3^2 = 9\). - **Conclusion**: \(2 < \sqrt{8} < 3\). --- **Summary of Results:** a) \(7 < \sqrt{56} < 8\) b) \(-4 < -\sqrt{12} < -3\) c) \(-9 < -\sqrt{78} < -8\) d) \(3 < \sqrt{15} < 4\) e) \(-7 < -\sqrt{43} < -6\) f) \(-6 < -\sqrt{29} < -5\) g) \(9 < \sqrt{99} < 10\) h) \(2 < \sqrt{8} < 3\)