THEORY OF EXPONENTS 285. By the ordinary definition of a power, a" is defined as the product of m factors each equal to a. In this restricted sense there is no first power of a, because. that is not the product of any number of a's, nor for a similar reason is there any zero power, negative power, or fractional power. 286. From the preceding it is evident that the original definition does not include expressions like a', ao, a3, a 3, a ̄1⁄2, a a-", etc., and hence these are as yet meaningless, except as meanings have been given to a' and ao in previous discussions. POSITIVE INTEGRAL EXPONENTS 287. Before proceeding to discuss the meanings which should attach to negative and fractional exponents, it will be an ad vantage to restate the four fundamental laws which have already been proved for exponents that are positive integers. These are: 288. Meaning of the Unit Exponent. We have already defined a1 to mean a (Art. 19). That this is the natural definition is seen from the fact that, 289. Meaning of the Zero Exponent. We have also seen that the expression ao may be defined to This meaning may also be derived as mean 1 (Art. 125). follows: Assuming a" × a" = am+" for all values of m and n, we have, That is, any number with zero exponent is equal to 1. Why must a0 in this form? May a be either positive or negative? NEGATIVE AND FRACTIONAL EXPONENTS 290. Since negative and fractional exponents are as yet meaningless, we assume the existence of Law I, Art. 287, for any rational values of the exponents, and derive consistent definitions. 291. Meaning of a Negative Integral Exponent. I. When the exponent is - 1. Let am × a" = am+", m and n being any integers. - 1, n = 2, m+n Since the assumption of a single law, viz., a" × a" = a"+", 1 yields a "= by a strictly logical process, we are justified in an making a definition for a negative exponent as follows: A Negative Exponent is a mode of expressing the reciprocal of a number with the same positive exponent. One practical application of this definition is that the value of a fraction is not changed if any factor be removed from numerator to denominator, or from denominator to numerator, and the sign of its exponent be changed. [Have the student give formal proof.] Assuming Law I, Art. 287, as true for any rational values of m and n, and hence true for rational fractional values of the exponents, let m = 29 By original definition of exponent (Art. 19), a2 × a2 = (a2)2. Hence (a)2= = a. (Axiom 1) With the assumption of a single law for fractional exponents we find by a logical process that the form a is a number whose square is a; hence we are justified in defining root of a, and we have the identity až as the square 1 II. Consider the form a", n being a positive integer. By original definition of an exponent (Art. 19), Hence we define a" to mean the nth root of a, and we have the identity 1 a" = Va. One more step is necessary for a complete knowledge of fractional exponents under the given assumption; viz.: m III. Consider the form a", m, n being positive integers. By assumption of Art. 287, I, for rational fractional exponents, and its extension to any number of factors, By extracting nth root of the members of the last identity, m a" = √a", m and with our assumption, we are justified in defining a" as the nth root of the mth power of ɑ, |