Determining Between Which Two Integers an Irrational Number Lies \( \begin{array}{lll}\text { Exercise 2: Without using a ealculator, determine between which two integers the following } \\ \text { irrational numbers lie: (Theory on paze5) } \\ \text { a) } \sqrt{28} \\ \begin{array}{lll}\text { b) } \sqrt{48} & \text { c) } \sqrt[3]{49} & \text { e) } \sqrt{90} \\ \text { autios: } 10 & \text { d) } \sqrt[3]{15} & \text { f) } \sqrt{115}\end{array}\end{array} \)

To find between which two integers an irrational number lies, you need to estimate the square root or cube root by identifying perfect squares or cubes nearby. For instance, \( \sqrt{28} \) is between \( \sqrt{25} = 5 \) and \( \sqrt{36} = 6 \), so it lies between 5 and 6. Similarly, for \( \sqrt{48} \), it's between \( \sqrt{49} = 7 \) and \( \sqrt{36} = 6 \), so it lies between 6 and 7. For \( \sqrt[3]{49} \), consider \( 3^3 = 27 \) and \( 4^3 = 64 \); thus, \( \sqrt[3]{49} \) is between 3 and 4. \( \sqrt{90} \) falls between \( \sqrt{81} = 9 \) and \( \sqrt{100} = 10 \), placing it between 9 and 10. The same approach works for the remaining numbers as well, rounding out a fuzzy yet fun exploration of irrational numbers!