Chapter 16: Exponent Rules: Situation 10 From the MACMTL-CPTM Situations Project

In an Algebra 2 class, students had just finished reviewing the rules for exponents. The teacher wrote x^m ⋅ xⁿ = x⁵ on the board and asked the students to make a list of values for m and n that made the statement true. After a few minutes, one student asked, “Can we write them all down? I keep thinking of more.”

The relevant mathematics in this Situation reaches beyond the basic rules for exponents into issues of the domains of the variables in those rules. The exponent rule x^m ⋅ xⁿ = x^m+n is applicable and is key to deciding how many solutions there will be. However, applying this rule beyond the usual context of positive bases and positive exponents to that of other number systems (such as the set of integers or rational numbers) requires consideration of the domains of the base and the exponents. In the following Foci, symbolic, numeric, and graphical representations are used to highlight that there are particular values in the domains of both x and m for which x^m is not a real number. In Focus 2 and Focus 3, the domains of x and m are extended and x^m is examined in these new domains.