71. The powers of the same quantity are multiplied together by adding the indices.

Thus, to multiply a2 by a3, it is necessary to write the letter a only once, and to give it for an exponent the sum 2+3, the exponents of the factors; that is, a2 × a3 == a2 +3 = a3, because a2 a×a, and a3 a Xaxa; therefore a2 × a3=a×a×a×a Xa=a. In general, the product of am by a", m and n being always entire positive numbers, is amin. In fact, am is the abbreviation of a Xa Xa,&c., continued to m factors, and a" is a XaXa, &c., continued to n factors; therefore a Xa"' = a xa xa x a Xa, &c., continued to m+n factors; which (Art. 12) is amin

Reciprocally amin can be replaced by aTM Xa". The quantity a" is sometimes called an exponential.

72. If two quantities having like signs are multiplied together, the sign of the product will be +; if their signs are unlike, the sign of the product will be.

I. A positive quantity being multiplied by a positive one, the product is positive; thus, a Xx + b = Fab, because +a is to be added to itself as often as there are units in b, and consequently the product will be +ab.

2. A negative quantity being multiplied by a positive one, the product is negative; thus, ax +bab; because -a is to be added to itself as often as there are units in b, and therefore the pro duct is ab. Or, since adding a negative quantity is equivalent to subtracting a positive one, the more of such quantities that are added the greater will the whole diminution be, and the sum of the whole. or the product, must be negative

3. A positive quantity being multiplied by a negative one, the product is negative; thus, +ax bab; because +a is to be subtracted as often as there are units in b, and consequently the product is -ab.

4. A negative quantity being multiplied by a negative one, the product is positive; thus, ax -b=+ab. For, ax-b-ab, that is, when the positive quantity a is multiplied by the negative quantity b, the product indicates that a must be subtracted as often as there are units in b; but when a is negative, its subtraction is equivalent to an equal positive quantity; therefore, in this case, an equal positive quantity must be added as often as there are units in b.

73. If all the terms of a compound quantity be multiplied separately by a simple one, the sum of all the products taken together, will be equal to the product of the whole compound quantity by the simple one.

For, in the first place, if a+b be multiplied by c, the product will be ca+cb: Since a+b is to be repeated as many times as there are units in b; the product of a by c, that is, ca, is too little by the product of b by c, that is, cb; it is necessary then to augment ca by cb, which will give for the product sought ca+cb, where the term +cb arises from multiplying +b by c. It would be found by reasoning in like manner, that the product of c by a +b must be ca+cb, where +cb is cx+b. If, in the second place, a-b be multiplied (where a is greater than b) by c, the product will be ca-cb. a-b is to be repeated as many times as there are units in c; the product of a by c will give too great a result by the product cb; it is necessary then to

Since

diminish the product ca by cb, so that the true product is ca-cb.

Let, for example, 7-2 be multiplied by 4; the product will be 28-8, or 20;

For, 7x4, or 23, is too great by 2x4, or by 8; therefore, the true product will be the first diminished by the second, or 28-8, that is 20. In fact, 7-2, or 5X4=20. The term cb of the product is the product of -b by c.

It would be found, by reasoning in like manner, that the product of e by a-b, must be ac-bc, the same as in the preceding, and in which the term --be is the product of c by-b.

If, in the third place, a+b+d be multiplied by c the product will be ca+cb+cd.

For, let a+b be designated by e; then, etd multiplied by c is equal to ce+cd; but ce is equal to cx (a+b)=ca+cb, because e is equal to a+b; therefore (a+b+d) Xe=ca+cb+cd. Also, if (a+b)-d be multiplied by c, the product will be ca+cb-cd; for let (a+b)=e, then (e-d) Xc=ce ~cd=c(a+b)—cd=ca+cb-cd.

Finally, it may be demonstrated, in like manner, that if any polynomial, a+b−d+e-ƒ, &c., be multiplied by c, the product will be ca+cb-cd+ce -cf, &c. Also, if a quantity c be multiplied by any polynomial a+b-d+e, &c., the product will be ac+bc-dc+ec, &c.

75. If a compound quantity be multiplied by a compound quantity, the product will be equal to every term of one factor, multiplied by every term of the other factor, and the products added together.

Let, in the first place, a+b be multiplied by c+d: a+b taken c times is ca+cb, as we have already proved; but this product is too little by the binomial a+b repeated d times, it is necessary then

to add to it da+db, and we will have ca+cb+da +db for the product sought; in which the term +db arises from the multiplication of +b by +d.

Suppose, in the second place, that a+b is multiplied by c-d, the product will be ca+cb-da-db. Because the product of a+b by c, that is, ca+cb, is too great by that of a+b by d, which is da+db ; we will have therefore the true product equal to ca+cb-da-db, where the term-db is the product of +b by d; in multiplying c-d by ab, we will find that bd is the product of d by +b.

Let, in the third place, a-b be multiplied by e-d; the product will be ca-cb-da-db.

For, the product of a-b by c, that is, ca-cb, is too little by that of a-b by d, which is da-db; because the multiplier c is too great by d; it is necessary then to subtract the second product from the first, and the difference will be (66) ca—cb—da+db.

Here the term +bd results from -b by ―d. Finally, if a+b+e be multiplied by c+d the product will be ca+cb+ce+ad+bd+de.

For, in designating a+b by h; then, (h+e)× (c+d)=hc+ec+dh+ed, which is equal hx (c+d) +ected=(a+b)x(c+d)+ected=ca+cb+ce +

ad+bd+de.

The same mode of reasoning may be extended to compound quantities composed of any number of terms whatever.

76. Cor. Hence, in general, if any two terms which are multiplied have different signs, their product must be preceded by the sign, and if they have the same sign, the product is affected with the sign; agreeably to what has been demonstrated (Art. 72,) where simple quantities, or isolated factors, such as, ta, tb, a, b, were only considered.

From the division of algebraic quantities into simple and compound, there arise three cases of Multiplication: the practical rules for performing the operation are easily deduced from the preceding propositions.

CASE I..

When the factors are both simple quantities.

RULE.

77. Multiply the co-efficients together, to the product subjoin the letters belonging to both the factors, and the result, with the proper sign prefixed, will be the product required.

Ex. 9. Required the product of 4abc and 3a2c.

Ans. 12a3bc2.

Ex. 10. Required the product of -7axy and -2acx.

3

Ans. 14a3cx3y.

Ex. 11. Required the product of 7x2y and -3y2x3.

Ans. 21xys.