Connecting rational exponents and functions

Georgia Journal of Science, 2002 by Reid, Denise, Petrella, Gerald, Goel, Sudmir

Abstract

An article by Evans and Tirosh, in the July 1995 issue of the Journal of Educational Studies in Mathematics, which addresses a difficulty with rational exponents is epitomized by the equation -2 = (-8)^sup 1/3^ =(8)^sup 2/6^ = [(-8)^sup 2^]^sup 1/6^ = 2. We suggest that perhaps a different perspective, namely a functional approach, is in order. We offer the reader a means of resolving the dilemma by defining the rational exponent as a function composition.

In the July 1995 issue of the Journal of Educational Studies in Mathematics, Evans and Tirosh claim that the quantity (-8)^sup 1/3^ is undefined. Based on their study, Evans and Tirosh argues that the vast majority of teachers participating in their study of "undefined mathematical operations," incorrectly argues that (-8)^sup 1/3^ = -2. They go on to suggest that (-8)^sup 1/3^ is undefined because

-2 = (-8)^sup 1/3^ = (-8)^sup 2/6^ = [(-8)^sup 2^]^sup 1/6^ = 2 (1)

If we take the argument given in equation (1) to its ultimate conclusion, it can be deduced that the negative real numbers do not exist. Let "a" be any positive real number. Then "-a" represents a negative number. According to the rationale suggested in equation (1),

-a = (-a)^sup 1^ = (-a)^sup 2/2^ = [(-a)^sup 2^]^sup 1/2^ = a.

Moreover, one can develop a parallel discussion to show that imaginary numbers do not exist. Evans and Tirosh have fallen into a common notational trap regarding rational exponents of real numbers. It is precisely because of such pitfalls that care must be taken when defining and manipulating such forms and why rational exponents should be revisited later in the curriculum. In 1997, Goel and Robillard rebutted the Evans and Tirosh arguments; using a classic definition of a^sup r^, where r is a rational number, it was shown that the quantity (-8)^sup 1/3^ is a well-defined real number with the value -2.

Key Words: rational exponents, mathematics education.

REFERENCES

1. Evans R and Tirosh D: Subject-matter knowledge and knowledge about students as source of teacher presentations of the subject matter. Educational Studies in Mathematics (An International Journal) 29: 1-19, 1995.

2. Goel S and Robillard S: The equation -2 = (-8)13 -= (-8)zib = [(-8)21 1,11 = 2. Educational Studies in Mathematics (An International Journal) 33: 319-320, 1997.

3. Sullivan M: "College Algebra," 5th edition, Prentice Hall, p72, 1999.

Denise Reid Gerald Petrella Sudmir Goel

Department of Mathematics and Computer Science Valdosta State University Valdosta, GA 31698 dtreid@valdosta.edu