Obferve, For either of thefe Kinds, viz. a mix'd Power, or mix'd Root, we may inftitute this manner of Representation A, which may fignify either the m Root of A", or the Power of A. But then obferve, that we can't make it reprefent either of these indifferently, till we have first demonftrated that they are equal; which fhall be afterwards done; and till then, I fhall only ufe it for the m Root of the " Power. n SCHOLIUM. Every Number is a Root of any Order whatever, because it may be involved to any Power; but every Number is not a Power of any Order; fome being Powers of no Order but the first, which is only being a Root; i. e. there are fome Numbers which cannot be produced by the continual Multiplication of any Number whatever; and fuch are 3, 5, 6, 7, and an infinite Number of others. Some again are Powers of one particular Order only; as 4, which is only a Square; and 8, which is only a Cube. Some, in the last place, are Powers of more than one Order, but limited to a certain Number of different Orders; as 64 is both a Square and a Cube; its Square Root being 8, and the Cube Root 4; for 8x8=4×4×464: The Demonftration of these things you'll learn afterwards; to mention them in general is enough here, which was only neceffary for the fake of the following Definition. n DEFIN. X. When a Number A is proposed as a Power of any Order ", and yet is not a Power of that Order, i. e. if it has no determinate Root of that Order, or there is no Number which involved as the Index " directs, will produce that Number; yet it has what we may call an indeterminate Root, (as fhall be afterwards explain'd) and this imagin'd Root, under the Notion of a true and compleat Root, is called a Surd (i.e. inexpreffible) Root, and is represented in the general Form A; and fuch Roots as are real, are, in Diftinction from Surds, called Rational Roots. Exam. 8 is not a Square; for there is no Number, which multiplied into itself, will produce 8. No Integer will, fince 2 X 24, and 3x3=9; and that no mix'd Number betwixt 2 and 3 can do it, will be afterwards demonftrated. But now as to Surds, don't mistake, as if fuch Roots or Representations were nothing at all, or fo merely imaginary as to be of no Ufe in Arithmetick; for though there be no fuch determinate or affignable Number, whofen Power is equal to A; yet we can find Numbers mix'd of Integers and Fractions, that fhall approach nearer and nearer to the Condition required, in infinitum; i. e. we can find a Series of Numbers decreafing continually, whofe Sum taken at every Step is a Number, the Power of which approaches nearer and nearer to the given Number; and this Series confider'd in its infinite Nature, as going on by the fame Tenor and Law without end, and thereby approaching infinitely near to the Condition of a true Root, is truly and properly what we call a Surd; which, 'tis plain, is fomething real in itself, though we can't exprefs the whole Value of it by any definite Number; for that is contrary to its Nature: So we find that the Surd Roots of different Numbers have certain Connections and relative Properties the fame way as rational Numbers have; (all which things fhall be demonftrated in their proper place.) Therefore we conceive Surds as Quantities compleat of their own kind, and so use the fame general Notation for Surds and rational Roots: And hence the following Theory relating to Roots are to be understood generally, whether they are Surds, or rational Roots; cor cerning the Reafon and Application of which to Surds, you'll learn more Particulars afterwards in Chap. 3. XI. The Powers and Roots of Fractions are to be understood the fame way as of whole Numbers; that is, any Fraction being continually multiplied into itself, is a Root or firft Power, with refpect to the Products which are the fuperior Powers. Obferve, That only is the proper and immediate Power of a Fraction, whofe Terms are the Powers of the Terms of that Fraction; yet as the fame Fraction may be expreffed in various Terms, fo all equivalent Fractions may be taken for the Power or Root of the fame Fraction, because they have the fame Effect in all Operations, if any one of them is fo, according to the Definition. Thus, because x, and because, therefore, which is the Square of, may be alfo called the Square of 4; alfo (=) may be cal led the Square of or . SCHOLIUM. When a complex literal Expreffion is confider'd as a Root, to exprefs any Power of it we draw a Line over the whole Expreffion, and then annex the Index; thus, A+B is the " Power of A+B, and AB is the Power of the Product AxB: But if there is no Line over, then the Index is applied only to the Member over which it is immediately fet, as A+B is only the Sum of A and B", and A Ba the Product of AxB. n AXIOM S. If, LIKE Powers or Roots of equal Numbers (however differently expreffed) are equal. Thus: If A=B, then A"=B", and AB But unequal Numbers have their fimilar Powers and Roots unequal; the greater having the greater Number for its Power or Root. Hence, COROL. If the different Powers of unequal Numbers are equal, the Power of the leffer Root has the greater Index. Thus: If A"=Bm, and A be lefs than B; then is " greater than m: For if nm, then is An less than Bm; and much more is it fo, if a is lefs than m. Exam. 824364. Ild, If a Number is involved to any Power, and from this Power a Root of the fame Denomination is extracted, this Root is the fame Number which was first involved. So if A is involved to the » Power, and of this Power, viz. A", we extract the Root, it is equal to A, i. e. A=A And reverfly extract any Root of a Number, and then involve that Root to a Power of the fame Denomination; this Power is the fame Number from which the Root was firft extracted; fo the " Power of A, or AA. Hence, n n COROL. I. An Expreffion of this Sort A, where the Numerator and Denominator of the Index are equal, whether it is understood as the " Root of the " Power; or the " Power of the Root of A, is no other thing in Effect and Value but A. Hence again; 2. Involution and Evolution are directly oppofite, the one undoing the Effect of the other; whereby they are mutual Proofs one of the other. THEOREM I. IF two fimilar Powers of different Numbers or Roots are multiplied together, the Product is the like Power of the Product of thefe Numbers or Roots. Thus, the Product of two Squares is the Square of the Product of their Roots. Universally Anx BAB. Demonftr. A=2, A2=4 AB 10. A× B2 Demonftr. In the Product Anx B", A and B are applied as Fa&tors equal times; fo that in the whole Product there is a Number of Factors equal to 2n; and confequently the Product A B, confider'd as one Number or Factor, is applied in that Product 2 times for in whatever Order the Factors are taken, the Product is still the fame]; fo that the Product given is AB raised to the Power, or AB. Thus particularly, A3x B3AAAX BBB = ABX ABX AB A B3. COROL. Hence we learn, how the Product of a Number expreffed as a Power, and another not expreffed as fuch (but fuppofed to be one) may be reduced to fuch an Expreffion. Thus, A" xB=Ax B; for the » Power of A is A", and the is B. therefore AnxB=Axb. = Power of Ba SCHOL. This Theorem is true alfo, when the Roots are the fame; as A"x A"= AA. But I have not taken this Cafe into the Theorem, because it falls within another (fee Theor. 6.) where the Product is expreffed in a more fimple and convenient way. to understand the fame of the Theorems 2, 3, and 4. THEOREM II. You are IF two fimilar Powers of different Numbers or Roots are divided one by the other, the Quote is the like Power of the Quote of the given Roots. Thus the Quote of two Squares is the Square of the Quote of their Roots univerfally Bn An=B÷÷A|"· A=3. A3 27 B=6 B=216 Demonftr. 1. Suppofe A, B are Integers, then BA expreffed fractionally is whofe » Power, according to Defin. 11. is B Α' n SCHOLIUM. I have not here confidered, whether B An is a whole Number or a Fraction; for which fo ever of them it be, you fee plainly that it is equal to B-A. In another place you'll find it demonftrated, that according as B"A" is integral or fractional, fo is BA; and reverfly. The following Corollaries 2, 3, 4, and 5, are deduced from this and the firft Theorem jointly confidered. COROLL. 1. The Quote of two Numbers, whereof one is expreffed as a Power and the other not, (tho' fuppofed to be one,) may be reduced to fuch an Expreffion n thus; A÷BA÷÷B; for An is the " Power of A, and B is the " Power of B. 2. If any Product and one of the Factors are fimilar Powers, the other is alfo a fimilar Power, and its Root is the Quote of the Roots of the other two Terms. Thus, if An Anx BD", then is BDAD-Al' whofe " Root is DA. In another Place it fhall be demonftrated that DA must be integral if DA or B is fo. III. If the Dividend and either of the other two Terms, viz. Divifor and Quote, are fimilar Powers, the other of those Terms is also a fimilar Power, whofe Root is the Product of the Roots of the Dividend and the former of the other two Terms. Thus if BAD", then A B x D"=BD". 4. The Product of two Numbers being a Power, the Factors are either both like Powers with the Product, or neither of them is fo. Alfo, if one Factor is a Power, and the other not a like Power, the Product is not a like Power. Let A B=p", if A is a Power of the Order ", fo is B, for if Aa" then is a" × B=p", and confequently B is of the Order (by the 2d) whence alfo if the one A or B is not a Power of the Order", neither is the other; for one being fo, the other would be fo too. Again, if A" x B =D, and B not a Power of the Order ", neither is D; for if D were fo, B would be fo too. 5. If the Quote of two Numbers is a Power, thefe Numbers are either both Powers like the Quote, or neither of them is fo: This is the Reverfe of the laft. SCHOLIUM. In thefe Corollaries you are to understand Rational Powers; for otherwife any Number may be reprefented as a Power of any Order. THEOREM III. If two fimilar Roots of different Numbers are multiplied, the Product is the like Root of the Product of these Numbers. Thus two Cube Roots produce a Number which is the Cube Root of the Product of their Cubes. Univerfally, Ax BAB“. Demonftration This is but the Reverfe of the 1ft Theorem, and follows easily from it. Thus, Let A B be any fimilar Powers, COROLL.. The Product of two Numbers, whereof one is expreffed as a Root and T the other not, may be reduced to fuch an Expreffion thus; A × B= A× B; for A is the Power of A", and B of B; therefore, by this Theorem Ax BAXB. This is alfo the Reverfe of Coroll. Theor. 1. THEOREM IV. IF two fimilar Roots of different Numbers are divided, one by the other, the Quote is the like Root of the Quote of the one Number divided by the other. Thus, two Cube Roots give for a Quote the Cube Root of the Quote of the Cubes. Univerfally, D÷A =DA 2. If D3, A are fractional, then are D, A alfo fractional, (according to a former Ob COROLL. The Quote of two Numbers whereof one is expreffed as a Root, and the other not, may be reduced to fuch, thus; D÷A=D÷A. For D is the Root of D, as A is of A". SCHOLIUM. From the two laft Theorems jointly confider'd, we have four Corollaries after the fame manner as the 2d, 3d, 4th, 5th Corollaries, deduced from Theorem Id and Ift. THEOREM V. THE Product or Quote of any like Mixt Power (or Root) of two different Numbers, is the like Mixt Power of the Product or Quote of thefe two Numbers: Thus = Demonftr. 1. Anx BAB" (Theor. I.) and Anx Ax Bol (Theor. III.) But AA and BB by the Notation; and fince A" x B = AB", hence Am xB==AB. 2. Aˆ÷÷B ® —A÷B” (Theor. II.) And A÷÷B = A÷Bam, (Theor. IV.) = n Example. The Square Root of 16 is 4, and the Cube of 4 is 64, therefore the Cube of the Square Root of 16, or 16 is 64. In like manner, the Cube of the Square Root of 81, or 81* is = 729. Then 81 × 16 = 1296, whofe Square Root is 36, and the Cube of this is 46656=64× 729; that is, 165 × = 16 x 81132. THEOREM VI. IF two Powers of the fame Root are multiplied, the Product is fuch a Power of the fame Root, whose Index is the Sum of the Indexes of the Factors. Thus, The Product of the 2d and 3d Powers of any Number, is the 5th Power of the fame Number. Univerfally, A" x A1=A"+". Example. A=3. A19 A 27. A2x A3 243. |