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Answers to Problem 4:

1.  a

Since a is between 0 and 1 it is a fraction.  When squaring a fraction the answer will become smaller than the original number.  An example of this:  If a = .5 then a² = .25 which is a smaller number.

2. 

The square of any positive or negative number will always result in a positive number, thus b² always will be bigger than the original number.  Example:  b = -.3

and b² = .09.

3.                           

Since 1/a is the reciprocal of the original number (and the original number is a fraction), the answer has to be larger because the answer always increase.  Example:  a = 1/3 then 1/a = 3.

4. 

Whenever you take roots of fractions, the value becomes larger each time you take a root.  The square root as a result should be smaller than the cube root which should be smaller than the 4^th root.  Example:  a = 1/16   the square root of a is ¼ and the 4^th root of a is ½.  Of these numbers, ½ is the larger value.

Students should be encouraged to explore the relationships between each of these situations to discover when each expression would be greater than the other one.  The domains for each of a and b are critical in affecting the correct relationship.  Using the other domains for the students to explore could generate extensions.