Final answer:
The roots given in the question can be placed between the ranges of 2 and 3, 4 and 5, or 9 and 10 by comparing them to known perfect squares and perfect cubes. Square roots and cube roots just under or over a known perfect square or cube indicate which interval they fall into.
Step-by-step explanation:
The student's task is to identify where each given root value falls between the ranges of 2 and 3, 4 and 5, or 9 and 10. We'll evaluate each expression to see which range they fall into:
- √101 ≈ 10 (since 102 = 100) so it is between 9 and 10.
- ³√90 is slightly less than ³√125 which is 5, so it falls between 4 and 5.
- ³√90, as stated above, falls between 4 and 5.
- ³√20 is slightly more than ³√8 which is 2, so it falls between 2 and 3.
- √20 is slightly less than √25 which is 5, so it falls between 4 and 5.
- √7 is slightly less than √9 which is 3, so it falls between 2 and 3.
- 8³√800 is hard to evaluate directly, but if you know that ³√512 is 8 and 800 is slightly larger, you could infer that 8³√800 falls between 9 and 10 since 8(2) = 16 which is between 9 and 10.
When determining the placement of these roots, it is useful to compare the values to perfect squares or perfect cubes that are known. Powers of two, three, four, etc., can be helpful guideposts. For cube roots, knowing that 23 = 8 and 33 = 27 can establish lower bounds while knowing that 43 = 64 and 53 = 125 can help establish upper bounds. For square roots, recognizing that 22 = 4 and 32 = 9 can help with the lower range whereas 42 = 16 and 52 = 25 help with the higher range.