2. Without the use of a calculator, deternine between which two integers the following irrational sumbers will lie, using a nember line: \( \begin{array}{lll}\text { a) } \sqrt{20} & \text { b) } \sqrt{11} & \text { c) } \sqrt{30} \\ \text { d) } \sqrt{200} & \text { e) }-\sqrt{8} & \text { f) } \sqrt[2]{70}\end{array} \)

To determine between which two integers the given square roots lie, first consider the perfect squares that are close to these numbers: - **a) \( \sqrt{20} \)**: The closest perfect squares are \( 4^2 = 16 \) and \( 5^2 = 25 \). Thus, \( \sqrt{20} \) lies between 4 and 5. - **b) \( \sqrt{11} \)**: The closest perfect squares are \( 3^2 = 9 \) and \( 4^2 = 16 \). Therefore, \( \sqrt{11} \) is between 3 and 4. - **c) \( \sqrt{30} \)**: The nearby perfect squares are \( 5^2 = 25 \) and \( 6^2 = 36 \). So, \( \sqrt{30} \) falls between 5 and 6. - **d) \( \sqrt{200} \)**: This lies between \( 14^2 = 196 \) and \( 15^2 = 225 \). Thus, \( \sqrt{200} \) is between 14 and 15. - **e) \( -\sqrt{8} \)**: Since \( \sqrt{8} \) is approximately 2.83, \( -\sqrt{8} \) lies between -3 and -2. - **f) \( \sqrt{70} \)**: The nearby perfect squares are \( 8^2 = 64 \) and \( 9^2 = 81 \), placing \( \sqrt{70} \) between 8 and 9. So the approximations are: a) 4 and 5 b) 3 and 4 c) 5 and 6 d) 14 and 15 e) -3 and -2 f) 8 and 9