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

Certainly! Let's determine between which two integers each of the given irrational numbers lies by analyzing their square roots. ### a) \( \sqrt{20} \) 1. **Find perfect squares around 20:** - \( 4^2 = 16 \) - \( 5^2 = 25 \) 2. **Determine the interval:** - \( 16 < 20 < 25 \) implies \( 4 < \sqrt{20} < 5 \) **Answer:** Between **4 and 5**. --- ### b) \( \sqrt{11} \) 1. **Find perfect squares around 11:** - \( 3^2 = 9 \) - \( 4^2 = 16 \) 2. **Determine the interval:** - \( 9 < 11 < 16 \) implies \( 3 < \sqrt{11} < 4 \) **Answer:** Between **3 and 4**. --- ### c) \( \sqrt{30} \) 1. **Find perfect squares around 30:** - \( 5^2 = 25 \) - \( 6^2 = 36 \) 2. **Determine the interval:** - \( 25 < 30 < 36 \) implies \( 5 < \sqrt{30} < 6 \) **Answer:** Between **5 and 6**. --- ### d) \( \sqrt{200} \) 1. **Simplify \( \sqrt{200} \):** - \( \sqrt{200} = \sqrt{100 \times 2} = 10\sqrt{2} \) 2. **Approximate \( \sqrt{2} \) (optional for better understanding):** - \( \sqrt{2} \approx 1.414 \), so \( 10\sqrt{2} \approx 14.14 \) 3. **Alternatively, find perfect squares around 200:** - \( 14^2 = 196 \) - \( 15^2 = 225 \) 4. **Determine the interval:** - \( 196 < 200 < 225 \) implies \( 14 < \sqrt{200} < 15 \) **Answer:** Between **14 and 15**. --- ### e) \( -\sqrt{8} \) 1. **Find perfect squares around 8:** - \( 2^2 = 4 \) - \( 3^2 = 9 \) 2. **Determine the positive interval:** - \( 4 < 8 < 9 \) implies \( 2 < \sqrt{8} < 3 \) 3. **Apply the negative sign:** - \( -\sqrt{8} \) lies between \( -3 \) and \( -2 \) **Answer:** Between **-3 and -2**. --- ### f) \( \sqrt[2]{70} \) (Assuming this means \( \sqrt{70} \)) 1. **Find perfect squares around 70:** - \( 8^2 = 64 \) - \( 9^2 = 81 \) 2. **Determine the interval:** - \( 64 < 70 < 81 \) implies \( 8 < \sqrt{70} < 9 \) **Answer:** Between **8 and 9**. --- ### Summary | Number | Interval Between Integers | |---------------|---------------------------| | \( \sqrt{20} \) | Between 4 and 5 | | \( \sqrt{11} \) | Between 3 and 4 | | \( \sqrt{30} \) | Between 5 and 6 | | \( \sqrt{200} \) | Between 14 and 15 | | \( -\sqrt{8} \) | Between -3 and -2 | | \( \sqrt{70} \) | Between 8 and 9 | By identifying the nearest perfect squares, we've determined the intervals where each irrational number lies on the number line.