(a) Without using a calculator, determine between whe inrational numbers lie. Then verify your answers by using a calc \( \begin{array}{lll}\text { (1) } \sqrt{50} & \text { (2) } \sqrt{29} \\ \text { (5) } 1 \sqrt[5]{30} & \text { (6) } \pi^{1}\end{array} \)

To find where the numbers lie between two rational numbers, we can approximate the square roots and the fifth root. For \(\sqrt{50}\), we know it lies between \(\sqrt{49} = 7\) and \(\sqrt{64} = 8\), so \(7 < \sqrt{50} < 8\). Similarly, \(\sqrt{29}\) is between \(\sqrt{25} = 5\) and \(\sqrt{36} = 6\), giving us \(5 < \sqrt{29} < 6\). Now, for \(1\sqrt[5]{30}\), observe that \(2^5 = 32\) and \(1^5 = 1\), hence \(1 < \sqrt[5]{30} < 2\). Lastly, \(\pi \approx 3.14\) shows that \(3 < \pi < 4\). Now let’s verify using a calculator. Calculating: 1. \(\sqrt{50} \approx 7.07\) which is between 7 and 8. 2. \(\sqrt{29} \approx 5.39\) which is between 5 and 6. 3. \(\sqrt[5]{30} \approx 2.24\) which is between 1 and 2. 4. \(\pi \approx 3.14\) which is between 3 and 4. All approximations confirm our placements!