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 86 ALGEBRA.ence (viz. unit), and that in the laft terms it is never abx c'—a+bx Ar c = «-fi £ ' + 2 cxX found. The powers of b are in the contrary order; it — a --2ab--b Ar2ac- c2bc--c. is not found in the firft term, but its exponent in the fe- In thefe examples, a Ar b c is confidered as comcond term is unit, in the third term its exponent is 2 ; pofed of the compound part aArb and the fimple part c; and thus its exponent increafes, till in the laft term it and then the powers of * ft- £ are formed by the4 precebecomes equal to the exponent of the power required.” ding rules, and fubftituted for aArb1 and <T+i . As the exponents of a thus decreafe, and at the fame time thofe of b increafe, “ the fum of their “ exponents is always the fame, and is equal to the Chap. VII. 0/* Ev glut ion. “ exponent of the power required.” Thus, in the 6th power of a b, viz. a6 + 6 b 15 «4 bx reverfe of involution, or the refolving of powers + 20 tf3 b* + 15 a4 £4 + 6 a £s + £ 6, the exponents intoThe their roots, is called evolution. The roots of fingle of a decreafe in this order, 6, 5, 4, 3, 2, 1, 0 ; and quantities extra&ed “ by dividing their expothofe of b increafe in the contrary order, o, 1,2, 3, 4, “ nents byaretheeafily that denominates the root re5, 6. And the fum of their exponents in any term is “ quired.” Thus,number the root of a8 is ai = a4 ; and 4 8fquare always 6. root of1a b cx is «4 b4 c. The cube root of 3 To find the coefficient of any term, the coefficient of the% £fquare i; and the cube root of x9 y6 s14 is the preceding term being known, you are to “ divide the x y'1 isz4. The= aground this rule is obvious from the <cV coefficient of the preceding term by the exponent of rule-for involution. Theofpowers any root are found b in the given term, and to multiply the quotient by multiplying its exponent by the ofindex that denominates “ the exponent of a in the fame term, increafed by u- by power ; and therefore, when any power is given, the “ nit.” Thus to find the coefficients of the terms of the root muft be found by dividing the exponent of the given the 6th power of a + you find the terms are power by the number that denominates the kind of root a6, a5 b, a4 bx, a3 P, ax b «bs, b6 ; required. and you know the coefficient of the firft term is unit; thatIt isappears, from what was faid of involution, that therefore, according to the rule, the coefficient of the “ any power that has a pofitive fign may have either a fecond term will be — X 5 + 1 = 6; that of the third “ pofitive or negative root, if the root is denominated “ by any even number.” Thusa the fquare-root of A~a 1 becaufe A~ X+^ or —aX—a gives term will be —X 4ft-x = 3X 5 = 15; that of the may2 befor theor'—a, product. But if a power have a negative fign, “ no root of it fourth term will be—X34-1 = 5X4 = 20; and “ denominated by an even number can be be afligned,” there is no quantity that multiplied into itfelf an thofe of the following will be 15, 6, 1, agreeable to the fince even number of times4can give a negative produtt. Thus preceding table. In general, if a + £ is to be raifed to anym power m m, thefquare root of—« cannot be affigned, and is what we the terms, without m 3 5 their m 4coefficients 4 m % swill be a, a —'b, call an impojjible or imaginary quantity. But if the root to be extrafted is denominated by an am—ib a A, a — A , — b , &c. continued till odd number, then ftiall “ the fign of the root be the fame the exponent of b becomes equal to vi. as the fign of the given number whofe root is required.” The coefficients of the refpe&ive terms, according to Thus the cube root of'—«5 is •—ay and the cube root of the laft rtile, will be I, m. 'i X --, X 3 If the number that denominates the root required is a divifor of the exponent of the given power, then ftiall the -3 1 X — x ' <-=2x ^=4, root be only a11lo-iuer power of the fame quantity. As the root of a is a4, the number 3 that denominates &c. 3.4 continued until you 2have one3 coefficient4 more .5than cube the cube roof being a divifor of 12. there are units in m. But if the number that denominates what fort of root Itmfollows therefore by thefe laft rules, that a bm1 is required is not a divifor of the exponent of the given m “ then the root required (hall have a fraction for — a + m a —b -f vi X —— 4-w X——2 power, 2 X its exponent.” Thus thefquare root of a3 is a-, the cube m 3 3 root of as is rf-f* and the fquare root of a itfelf is a. X —ix a i + X X X X Thefe powers that have fraftional exponents are called X am3—4 b* +, &c. which is 234 the general theorem for imperfect poolers or furds; and are otherwife exprefied railing a quantity confifting of two terms to any power »/. by placing the given power within the radical fign /, If a quantity confifting of three or more terms is to be and placing above the radical fign the number that denoinvolved, “ you may diftinguifti it into two parts, confi- minates what kind of root is required. Thus dering it as a binomial, and raife it to any power by the preceding rules; and then, by the fame rules, you a^—n/a* ; and a7—/am. In numbers the fquare root may fubftitute, inftead of the pdwers of thefe compound parts, their values.” Thus, of 2 is exprefled by 4/ 2, and the cube root of 4 by 4/ 4* The