PROOF OF THE INDEX LAWS FOR NEGATIVE, FRACTIONAL, AND NEGATIVE AND FRACTIONAL VALUES OF m and n

245. In the following proofs, m and n are rational integers or rational fractions.

The Law aTM× a" = am+n ̧

1. When m and n are negative and integral.

2. When m and n are positive and fractional.

Let m= P and n =

P

ps

qr

аяха =ags × a 9a = apsar /aps x air = Vaps+gr = a

=

3. When m and n are negative and fractional.

ps+qr 98

P

Let the student discuss this law when m is a positive and n a negative fraction.

The Law (am)" = amn.

1. When m and n are negative and integral.

Let m =- - p and n = - q, p and q being positive and integral.

mp m.P

(am)n = (am) a = √(am)p = √amv = a ̈a = aTM·o ̧

Let the student discuss this law (1) when m and n are both positive and fractional, and (2) when m and n are both negative and fractional.

APPLICATIONS OF THE PRINCIPLES OF EXPONENTS

(a) SIMPLE FORMS INVOLVING INTEGRAL EXPONENTS 246. In processes with exponents no particular order of method can be said to apply generally. Experience with different types will familiarize the student with those steps that ordinarily produce the clearest and best solutions. As a rule, results are considered in their simplest form when written with positive exponents.

(b) TYPES INVOLVING THE FRACTIONAL FORM

248. In the following illustrations attention is called to each important feature of the process, and the order of the principles that is emphasized in each is such as will, under similar conditions, produce the best form of solution.

Illustrations:

1. Simplify (a2b ̄31⁄2−1) -2.

(a2b ̄}x−1)−2 = a−b}x2 =

=

biz2. Result.

a1

Note (1) that the first step is the application of the law (am)n = amn, and (2) that the result is given with all exponents positive.

2. Simplify (c2 √c ̄1)3.

(c2 √c−1)8 = (c2. c ̄1)8 = (c12)2 = c2.

Result.

Note that the law am × an = am+n is first applied so as to unite c-factors

3. Simplify {√(Va--+ {√(√a-3)-1}

{√(√a ̄®)-} } ~} = {√(a ̄})} } + = {√π} = {a} } + = a-3 = 1.

Note that the reduction is accomplished outward.

Result.

_251 m2 125 m2

25 √m

27

Result.

Note that in the third step inverting the fraction changes the sign of

the exponent of the fraction. In general,