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: 3^{2} means two factors of three. By order of operations,( 3^{2} )^{5 } means 5 factors of 3^{2 }. Next, 3^{10} / 3^{2} reduces to 3^{8} , 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 16

^{1/4 }as 1 of four factors of 16. That makes perfect sense to me. And 16

^{1/3}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.