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 $a$, $a^{frac{m}{n}}$ is defined as $a^{frac{m}{n}}={^n}sqrt{a^m}$, where $m{in}mathbb{Z}$ and $n{in}mathbb{N}$ have common prime factors. For that situation, the author shows his opinion that the definition is not well-defined and $a^{frac{m}{n}}$ must be defined as $a^{frac{m}{n}}=({^n}sqrt{a})^m$, whenever $^nsqrt{a}$ 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: $${a^{r+s}=a^r{cdot}a^s\(a^r)^s=a^{rs}\(ab)^r=a^rb^r$$ for $a$, $b$>0 and $s{in}mathbb{R}$ 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.