tiply the co-efficients 8 and 3, giving 24 for the co-efficient of the product, and write down the letters of the multiplicand and multiplicator, prefixing the sign+, as the fac tors have like signs. + 8am Again, when the factors are compound quantities, each term of the multiplicand is to be multiplied by each term of the multiplicator; and the sum of these several products, collected according to the rules for addition, will give the product required. Examples. Prod. an+mn — nx — ac- cm + cx+az+mz−x Z In the first of these examples, 5 a multiplied by 2 a, give 10 aa; and both having the like sign+, the product has also that sign; in the same way m multiplied by 2a, gives 2 am, also a positive quantity; but the third term, — 2c, multiplied by + 2a, gives 4 a c, with the sign the signs of the factors are unlike. because In the second example, where the multiplicator consists of two terms, a multiplied by x, gives xx; then z multiplied by x, gives xx; again working with z, the second term of the multiplicator, we have zx, or more properly xz, (in order to place the letters as they stand in the alphabet, for the value is the same,) which is written under the second term, and z by equal to zz: then summing up these two lines of product, beginning either at the right or the left hand, we have xx + 2 x z + zz for the product required. In the third example, the characters of the multiplicand and multiplicator being all different, none of the products obtained by working with the several terms of the multiplicator can be combined together; consequently the three lines of product must be written down successively in one line, as was directed to be done in performing addition. These operations may be illustrated by using arithmetical numbers; take, for instance, the second, where + is to be multiplied into itself or by x+x. Let x stand for 4, and z for 6; then the operation would appear in the following shape: Here the component parts of the given quantity are multiplied separately together, and the result is the sum of the product of 4 by 4, of twice 4 by 6, and of 6 by 6, making 100, which is equal to the product of the whole of these component parts, or 10 multiplied into itself. In this example, in summing up the separate products, we find+am and -a m, which destroy each other; that is, if the one be subtracted from the other, as must be done, since they have unlike signs, there will be no remainder; a point, therefore, or asterisk, is placed in the general product: and in the same manner the two quantities-ax and +ax destroy each other, on which account another asterisk is placed in the product, and the following terms are successively brought down. To illustrate this example by common arithmetic, let a represent 8, 6, and x 4, then the question will be thus performed. 8+6-4 64 -36 The products arising from the repeated multiplication of a quantity into itself, and its several products, are called its powers; and the quantity itself is the root; thus a is the root, and if multiplied into itself, the product, or second power, will be aa; which again multiplied by a, will give the third power, or aaa; the 4th power will be aaaa, and so on; but as this manner of expression would soon become inconvenient, and liable to error, it has been the practice to express the different powers by a small figure placed over and towards the right hand of the root; thus the 2d power, aa, may be expressed by a2, the 3d power aaa by a', the 4th power by at, &c. This small figure is called the exponent, or index; and, by adding these exponents, the same expression is obtained as if the root had been repeatedly multiplied; thus a + a2 = a3, and a2 + a4 = a, &c. Division of algebraic quantities is performed agreeably to the following rules. When the signs of the divisor and the dividend are like, the sign of the quotient is +; but if they be unlike, the sign is -. 1st. When the divisor is simple, and a part, of or found in each term of the dividend, you must divide the co-efficient of each term of the dividend by the co-efficient of the divisor, and expunge, or withdraw from each term, the letter or letters of the divisor, and the result will be the the quotient. Thus, if it be required to divide 18mx by 3m, dividing the co-efficient 18 by 3, we have 6 for the co-efficient of the quotient; and mx being a product of which m is one of the factors, this symbol being taken away or expunged, the remaining symbol x will belong to the quotient, which will then be 6x. Again divide 5 a3m + 25 abm-5 am2 by 5am, and the quotient will be a +5b-m, thus: bam 5 am) 5 a3 m + 25 al m-5 am2 (a2 + 5 b—m In beginning this operation, the divisor 5am, is to be taken out of 5a3 m; if we take the root a from the 3d power of a, we have the 2d power of a, which therefore goes into the quotient; and as 5 is contained once in 5, and m once in m, neither this co-efficient 1, nor the quotient 1, are required to appear; they are, therefore, suppressed or expunged, and the divisor, 5 am, being multiplied by the quotient a', the product written under the dividend, and subtracted from it, gives no remainder. The next term of the dividend, +25 a b m, is brought down, and the 5 of the divisor being contained 5 times in the 25 of this dividend, 5 is placed as a co-efficient in the quotient; then a m is contained & times in ab m, and 5b becomes the quotient corresponding to this step of the division; which quotient multiplied into the divisor, gives 25 a bm to be subtracted from the dividend. Lastly, comes down the term — 5 a m2, out of which, if we take the divisor + 5 am, the quotient will be m2; so that the whole quotient in this operation will be a 5 b — m, Or, the same operation may be shortened by expunging the characters of the divisor from the dividend as below. 5 am)5 a3 m + 25 a b m — 5 am2( |