What is So Negative About Negative Exponents?

Abstract

While teaching college-level mathematics (from College Algebra to Calculus to Abstract Algebra), I have observed that students are often uncomfortable using negative exponents in calculations. I believe the fault partially lies in the manner in which negative exponents are taught in Algebra 1 or Algebra 2 courses, especially in rigid instructions always to write answers using only positive exponents. After reviewing a sample of algebra texts used in the United States over the last two centuries, it appears that while attitudes toward negative exponents have varied from author to author over time, the current trend is to declare explicitly that an expression is not simplified if it contains negative exponents. I believe that this negative attitude toward negative exponents is at least somewhat to blame for students of Calculus and higher mathematics being less able to solve problems that require conversion between positive and negative exponents, as their algebraic instruction has only taught them to convert negative exponents to positive.