Final answer:
To approximate and order √24 and √8, we find the nearest perfect squares and estimate the square roots as 5 and 3 respectively. We then compare these with the perfect squares to establish the order from least to greatest.
Step-by-step explanation:
The student's question pertains to the comparison and ordering of real numbers and finding rational approximations of square roots using perfect squares. To address this question, let's explore a mathematical approach for approximating square roots like √24 and √8. We compare and order these square roots by identifying the nearest perfect squares.
For instance, the perfect squares nearest to 24 are 16 (4²) and 25 (5²). Since 24 is closer to 25 than to 16, we say that √24 is approximately 5. Similarly, the perfect squares closest to 8 are 4 (2²) and 9 (3²). Since 8 is closer to 9, we approximate √8 as 3. Therefore, the order from least to greatest is 2 < 3 ≈ √8 < 4 < 5 ≈ √24 < 5.
Approximation, sometimes, may rely on a degree of conceptual understanding rather than strict calculative precision. Rational approximations and comparisons are used in various mathematical contexts, including equilibrium problems and analyzing numbers in scientific notation to determine their order of magnitude. Appreciating the closeness of numbers rather than their exact value can be a useful skill in problem-solving.