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 R A. B A L G E 82 a + b) a*2 + 2 a 5 + i2 (a + £ root, to the right hand, a figure exprefling the number a +aA of faftors that produce them. Thus, a v. J3 f ift Power of the^-^ a b + b' aa r-a S2C^ r root<?, andVa* a b b% aaa V^Kgdv is (hortly a a a a a X S A4thX expreflfed C 5* o . o. aaaa ^ yth-^ thus, ' a the firft term of this new dividend by the Thefe figures which exprefs the number of faftors that “ “firftDivide term of the divifor, down the quotient, produce powers, are called their indices or exponents; (which in this example isand£), fetwith’ its proper fign. thus 2 is the index of a2. And powers of the fame2 root3 ““ Then multiply the whole divifor by-this part of the are multiplied by adding their exponents. Thus a X a “ quotient, and fubtradl the produdt from the new diviand if there is no remainder, the divifion is fiSometimes it is ufeful not actually to multiply com- ““ dend; niffied:” If there is a remainder, you are to proceed pound quantities, but to fet them down with the fign of after the fame manner, till no remainder is left; or till multiplication (X) between them, drawing a line over it appear that there will be always fome remainder. each of the compound factors. Thus a + £ X a — b Some examples illuftrate this operation. exprefles the product of a + multiplied by a — b. Examp. I. a + will £) a21 — i2 (a — b a ab Chap. IV. 0/* Division. — ab — bx •— a b — b* The fame rule for the figns is to be obferved in divifion as in multiplication ; that is, “ If the figns of the o. o. dividend and divifor are like, the fign of the quotient “ muft be +; if they are unlike, the fign of the quo- Examp. II. a—3) aaa—3aa3+3a33—333(aa—2alfbb “ tient mult be —.” This will be eafily deduced from aaa— aab the rule in multiplication, if you coufider, that the quotient mult be fuch a quantity as, multiplied by the divi—2aa3-j-3a33—333 for, fhall give the dividend. —2aa3+2a33 The general rule in divifion is, “ to place the dividend “ above a fmall line, and the divifor under it, expunabb—bbb “ ging any letters that may be found in all the quantiabb—bbb “ ties of the dividend and divifor, and dividing the co“ efficients of all the terms by any common meafure.” o. o. Thus, when you divide o a b 15 a csy io a d, ex- It often happens, that the operation may be continued punging a out of all the teims, and dividing all the without end, and then you have an infinite for the and quotient; and by comparing the firjl three orferies d coefficients by 5, the quotient is 2i+i£ four terms 4 you may find what law the terms obferve: by which means, without any more divifion, you may continue the as far as you pleafe. Thus, in dividing 1 by “ Powers of the fame root are divided by fubtrafting quotient j — a, you find the quotient to be i+a+aa-j-aaa “ their exponents, as they are multiplied % 2 by adding a a a a + &c. which feries can be continued as far as “ them.” Thus, if you divide a by /*, the6 quotient the powers of a. is a5-27 or a3. And b26 divided by5 2 gives b —* or £2; you pleafe, by adding l The operation is thus : and a V divided by a b gives a £ for the quotient. “ If the quantity to be divided is compound, then l — a) 1 (ift-tf + aa-f-aaa,. “ you mull range fts parts according to the dimenfions of 1 —a “ fome one of its 2letters, as in the2 following example.” In the dividend a + 2 a A -p £, they are ranged ac2 +a cording to the dimenfions of a, the quantity a, where + a—a a a is of two dimenfions, being placedl firft, 2 a b, where it is of one diinenfion, next, and b, where a is not at aa all, being placed lall. “ The divifor muft be ranged + a a — aaa “ accorcfing to the dimenfions of the fame letters; then “ you are to divide the firft term of the dividend by the +aaa ““ firft divifor, isanda;tothen fet down the this quotient, ft-aaa — a a a a which,termin ofthistheexample, multiply quo“ tient by the whole divifor, and fubtratt the product a a a a, &e. “ from the dividend, and the remainder (hall give1 a new Note, The fign -f- placed between+any two quanti“ dividend, which, in this example, is a b + b ties,