Comments On the Definition of the Rational Exponent $a^{\frac{m}{n}}$ in Contemporary Korean Highschool Mathematics Textbooks

고등학교 수학 교과서에 제시된 유리수 지수 $a^{\frac{m}{n}}$의 정의에 관한 소고

There may be two methods defining the rational exponent $a^{\frac{m}{n}}$ for any positive real number a. The one which is used in all korean highschool mathematics textbooks is to define it as $\sqrt[n]{a^m}$, that is $(a^m)^{\frac{1}{n}}$. The other is to define it as $(\sqrt[n]a)^m}$, that is $(a^{\frac{1}{n}})^m$. In this paper, we insist that the latter is more appropriate and universal, and that the contents of current textbooks on the definition of the rational exponent should be corrected.