What is a teacher (or student) supposed to do if a rebellious exponent breaks a law or rule? Shall we put it in the corner? Assign a detention? Every time I read about “laws” and “rules” for exponents I don’t have to curl my hair for 3 days because somehow we math teachers need to get the word across: Math is not defined as memorized sets of steps to follow. If my student asks me, “Do I add these or do I multiply,” I have failed to ingrain the definition of an exponent and properties that distinguish an exponent from a coefficient.
The Common Core State Standards for Math (CCSS-M) include a Real Number System N-RN high school cluster, “Extend the properties of exponents to rational exponents.” That cluster does not include a list of rules to follow (add here, subtract there, multiply for this, and cry over that). The more I study the CCSS-M, the more I appreciate the intentional wording and exacting detail. Because my math background is mostly with traditional textbooks, I am still figuring out those nuances, so you may have noticed this one long before I did. But for those who didn’t, let me explain.
Exponents prescribe a number of factors: 32 means two factors of three. By order of operations,( 32 )5 means 5 factors of 32 . Next, 310 / 32 reduces to 38 , but not because the exponents are trying to avoid punishment. Rather, 3/3=1.
Understanding the definition of an exponent makes life so much easier. For example, I now see 161/4 as 1 of four factors of 16. That makes perfect sense to me. And 161/3 makes no sense at all because 16 doesn’t have 3 factors of anything so it is irrational. I can even have fun with the idea of rationally square rooting a decimal number: (6.25)1/2 is not irrational because 6.25 has two factors of 2.5.
A great step in convincing Americans that math beyond basic arithmetic is useful, would be to take the focus off step-by-step memorized procedure and endless lists of laws and formulas. Let’s keep exponents out of our court system and clearly define them instead.