CHAPTER XIX

THEORY OF EXPONENTS

239. The Index Laws for Positive Integral Values of m and

240. The extension of the practice of algebra requires that these laws be also extended to include values of m and n other than positive integral values only; and the purpose of this chapter is to so extend those laws. We shall, therefore:

1. Assume that the first index law (a" × a" = am+") is true for all values of m and n.

2. Define the meaning of the new forms that result under this assumption, m or n, or both, being negative or fractional.

3. Show that the laws already established still hold true with our new and broader values for m and n.

THE ZERO EXPONENT

If m and n may have any values, let n=0.

241. Any quantity with the exponent 0 equals 1.

If m and n may have any values, let n be less than 0 and

Hence, we define am as 1 divided by am.

From this definition we obtain an important principle of constant use in practice:

242. Any factor of the numerator of a fraction may be transferred to the denominator, or any factor of the denominator may be transferred to the numerator, if the sign of the exponent of the transferred factor is changed.

3. 2 ab-2. 6. 3 a 3x2y1. 9. 3-1a-2b3c-1.

Read the following without denominators:

12. 3-1a-ma1y.

If the expression a can be shown to conform to the first index law, we may find a definition for the fractional form of exponents. By the first index law,

(a1)2 = a‡ × a1 = a1+1 = a.

Hence the meaning of a is established, and the exponent in this form still agrees with the fundamental index law. That is:

a = √a is one of the two equal factors of a.

Similarly, (a) = aš × a3 × a} = aŝ+ŝ+j = a§ = a2. a3 is one of the three equal factors of a2.

That is,
In like manner, a

Therefore:

a = Va
a = Va3

= one of the three equal factors of a.
three of the four equal factors of a.

243. In the fractional form of an exponent we may define the denominator as indicating a required root, and the numerator as indicating a required power.

Illustration:

V82 = 8 = 22 = 4. And 4 contains two of the three equal factors of 8. √/813 = 81a=33=27. And 27 contains three of the four equal factors of 81. In general:

That is, a" is an expression whose nth power is am.

Or, as above, a" is one of the equal factors of a".

It is understood that while a must equal either + √a or – √a, we consider the positive value only; and a is defined as +√a, the principal square root of a.

In future operations we may apply the definition of Art. 243 to expressions given in radical forms, observing that

244. The index of a radical may be made the denominator of an exponent in the fractional form, the given exponent of the power of the quantity becoming the numerator of the fractional form

And in the converse operation of the principle of Art. 244:

Having established a meaning for the new forms of ex

m

ponents, ao, a-1, and an, we must show that the index laws hold true for these new forms; thus fulfilling the third and final clause of our agreement in Art. 239.

SOM. EL. ALG.-15