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 98 A L G E B R “ them by one another, andfet the common radical fign dufts. But when any compound furd is propofed, there 5 := is another compound furd 'which multiplied iprfjit gives “ over their product or quotient.” Thus, Xy'Zi a rational product. Thus, /fa-Arnfb multiplied by ifa —/A a—by and inveftigation “ whichgives multiplied into “thethepropofed furdofwillthatgivefurda “ rational product,” is made eafy by the following the5 If the furds have not the fame radical fign, “ reduce ‘ them to fuch as (hall have the fame radical fign, and THEOREMm I. m Generally, if you multiply a —b, gby a”continu— m fm n nm m a/ a CCi “ proceed as before,” ^aX/b—/a’'b ; » ~ an—xmjjm _^.an—imfoim V x ed till the terms be in number equal to —, the pron n r ' a ’n—bm V'8xi6=a/i28. If the furds have any rational coeffi- r n cients, their prcdudf; or quotient muft be'prefixed; thus, a +an —mbtn+ann—zx«‘b1 w+a,&c. —a —mbm.—a — mb'm—*»—3m£3,„, &c<—yn 2a/~io/18. The powers of furds are found as the powers of other quantities, “ by multiplying their exponents by the in“ dex of the power required ;”'thus the fquare of a/2 is THEOREM II. 2t 2 ~2t =y^4i ^ctibeof a/5—^=5:i'=a/i2S. ati—m—an—tmbm+an—lmbim.— an~*mbtmt &c. mufn m Or you need only, in involving furds, “ raife the quan- tiplied by a> --b gives a’‘z+zbn> which is demonilrated « tity under the radical fign to the power required, con- as the other. Here the fign of b» is pofitive, when — “ tinuing the fame radical fign ; unlefs the index of that “ power is equal to the name of the furd, or a multiple odd number. “ of it, and in tfiat-cafe the power of the-furd becomes is anWhen furd is propofed, “ fuppofe the “ rational.” Evolution is performed “ by dividing-the “ index ofanyeachbinomial number equal to m, and let n be the “ fraftion which is the exponent of the furd by the “ leaf! integer number that by m,a compound then Ihall “ name of the root required.” Thus the fquare root “ qn—m^tzan—*tiibm-.an~'i>nbis rmeafured ni, &c. give 4 . 3 z or e 4 3 “ furd, which multiplied into the propoftd furd a»r^=.bm of aAj is a/<3 a/<j. “ will givexa rational product.” Thus to find the furd 1 The furd A/tf” * ~ ; and in like manner, if a which multiplied by fa—fb, will give a rational quanpower of any quantity of the fame name with the furd tity. Here tu—i^ and the lead number which is meadivides the quantity-under the radical fign without a reby -f is unit ; let n—i, then IhsJl an— n in 1 mainder, as here am divides a,nx, and 2 5 the fquare of fured bhi+a —' b ™, 8cc. =aI~T-+-a1 } br+ac hr=ar-j-aSr 5 divides 75 the quantity under the fign in y/? 5 without which multiplied by fa— a remainder, then place the root of that power rationally before the fign, and the quotient under the fign, and thus fb gives a—b. the furd will be reduced sto a more Ample expreflion. To find the furd which multiplied by ffifffT— Thus, a/4 =V'3Xi6=4'v/3 i V'2i = a--b, givesna rational product. Here and n—.%, 3 a/75==5a/3; 3 and »—a —imbm+an—lmb*m} See. =a ^—a^~* V 27X3=3 y/ 3’ 9 When fords by the. lafl .article are reduced to their bl b^—al-l b%=„%-a*bl+aibZ-b%=f7 — leaft expreifions, if they have the fame irrational part, 9 they are added or fubtrafted, “ by adding or fubtradling “ their rational coefficients, and prefixing the fum or —ffbi+fJW—fT* “ difference to the common irrational part.” Thus, THEOREM1 III. / f / / A/75+v 3 :^: 48=5v 3 3+4V 3=9* 3; a/3 1+a/s4-34/3 'LetaOK±sbl be multiplied by at —piJf-«»—»m -^-2A/’ 5A/3•fards are fuch as confill of two or more Compound and the produdl fliall give an~d=:b™-l ; joined together. The fimple furds are commenfurable “h*l-ynt*—*mb-U. therefore » muft be taken the leaft integer that fnail in power, and by being multiplied into themfelves give at length rational quantities; yet compound furds multi- “ give — alfo an- integer. plied into themfelves commonly give itili irrational proDem.
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