The Meaning of the Extensions of Number Systems in School Mathematics and the Error Analysis Involved in the Interpretations of $(-8)^{\frac{1}{3}}$

$(-8)^{\frac{1}{3}}$에 내재된 수 체계 확장의 의미와 오류 해석

In this paper, we study the subject-matter knowledge related to the problem about rational exponent with negative bases. From the school mathematics point of view, we first investigate the meaning of the extensions of the number systems. We analyze the intrinsic meaning involved in the (-8)$^{1}$ 3) through the natural interpretation of rational exponent with negative bases by the complex number. we explain why it is important for a teacher to have the subject-matter knowledge in order to detect and correct student\`s mistake and misunderstanding.