Abstract
In all Mathematics I Textbooks(Kim, S. H., 2010; Kim, H. K., 2010; Yang, S. K., 2010; Woo, M. H., 2010; Woo, J. H., 2010; You, H. C., 2010; Youn, J. H., 2010; Lee, K. S., 2010; Lee, D. W., 2010; Lee, M. K., 2010; Lee, J. Y., 2010; Jung, S. K., 2010; Choi, Y. J., 2010; Huang, S. K., 2010; Huang, S. W., 2010) in high schools in Korea these days, it is written and taught that for a positive real number <TEX>$a$</TEX>, <TEX>$a^{\frac{m}{n}}$</TEX> is defined as <TEX>$a^{\frac{m}{n}}={^n}\sqrt{a^m}$</TEX>, where <TEX>$m{\in}\mathbb{Z}$</TEX> and <TEX>$n{\in}\mathbb{N}$</TEX> have common prime factors. For that situation, the author shows his opinion that the definition is not well-defined and <TEX>$a^{\frac{m}{n}}$</TEX> must be defined as <TEX>$a^{\frac{m}{n}}=({^n}\sqrt{a})^m$</TEX>, whenever <TEX>$^n\sqrt{a}$</TEX> is defined, based on the field axiom of the real number system including rational number system and natural number system. And he shows that the following laws of exponents for reals: <TEX>$$\{a^{r+s}=a^r{\cdot}a^s\\(a^r)^s=a^{rs}\\(ab)^r=a^rb^r$$</TEX> for <TEX>$a$</TEX>, <TEX>$b$</TEX>>0 and <TEX>$s{\in}\mathbb{R}$</TEX> hold by the completeness axiom of the real number system and the laws of exponents for natural numbers, integers, rational numbers and real numbers are logically equivalent.
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