From this it follows that any factor may be transferred from the numerator to the denominator of an expression, or vice-versâ, by merely changing the sign of the index.

/(27)2/3632-9

that am÷÷a1=am-n for all values of m and n.

241. The method of finding a meaning for a symbol, as explained in the preceding articles, deserves careful attention. The usual algebraical process is to make choice of symbols, give them meanings, and then prove the rules for their combination. Here the process is reversed; the symbols are given, and the law to which they are to conform, and from this the meanings of the symbols are determined.

242. The following examples will illustrate the different principles we have established.

9. 2x3.x-1. 10. 1÷2a. 11. x2xx1. 12. a2x1÷3x.

17. a ̄2x2 ́ ̄÷a3. 18. /a-1÷/a. 19. /a/a.

Express with radical signs and positive indices:

243. To prove that (am)namn is universally true for all values of m and n.

CASE I. Let n be a positive integer.

Now, whatever be the value of m

(am)n=am. am. am... to n factors

=am+m+m+... to n terms.

=amn

CASE II. Let m be unrestricted as before, and let n be a positive fraction. Replacing n by 2, where p and q are positive

Hence by taking the qth root of these equals,

CASE III. Let m be unrestricted as before, and let n be any Replacing n by -r, where r is positive, we

negative quantity.

have

Hence Prop. III., Art. 235, (am)n=amn has been shown to be

universally true.

244.

To prove that (ab)"=a"b", whatever be the value of n;

a and b being any quantities whatever.

CASE I. Let n be a positive integer.

=(a.a. a... to n factors) (b.b.b... to n factors) =anbn

CASE II. Let n be a positive fraction. Replacing n by

P

Taking the qth root, (ab)=aibi.

CASE III. Let n have any negative value. Replacing n by −r, where r is positive,

1

(ab)

= a¬rb-r

Hence the proposition is proved universally.

The result we have just proved may be expressed in a verbal form by saying that the index of a product may be distributed over its factors.

NOTE. An index is not distributive over the terms of an expres

sion.

Thus (a2+ b2)2 is not equal to a+b. Again (a2+b2) is equal to √a2+b2, and cannot be further simplified.

Examples. (1) (yz)a-°(zx)°(xy)—c=ya ̄c za ̄c z© x© x ̄©y ̃o,

=ya-2c za.

-kl

(2) {(a - b)}-x {(a+b)-*}=(a - b)-x (a+b)-kl

= {(a - b)(a+b)}-kt,
=(a2 — b2)-kl.

245. It should be observed that in the proof of Art. 244 the quantities a and b are wholly unrestricted, and may themselves involve indices.

14

(z2y-3)

2 2

2

Examples. (1) (x3y ̄3⁄43÷(x2y−1) ̄}=x3y ̃¦÷x ̄}y}

4

1

10. (x÷x)". 11. (x × *x n)i-n. 12. (Vo÷2x)1—a ̧