Whence it follows that in extracting the Roots of all compound Quantities, there muft be confidered, 1. How many different Letters (or Quantities) there are in the given Power. 2. Whether the single Powers of each of those Letters be of an equal Height, and have in them fuch a fingle Root as is required: which if they have, extract it as before. 3. Connect those single Roots together with the Sign +, and involve them to the fame Height with the given Power; that being done, compare the rew raised Power with the given Power; and if they are alike in all their respective Terms, then you have the Root required; or if they differ only in their Signs, the Root may be easily corrected with the Sign - as occafion requires. Example 1. Let it be required to extract the Square Root of cc+2cb-2cd + b b - 2 b d + dd. In this Compound Square, there are three diftinct Powers, viz. bb, cc, dd, whose single Roots are b, c, d, wherefore I suppose the Root, fought to be b + c + d, or rather bc-d, because in the given Power there is - 2 cd, and - 2bd, therefore I conclude it is -d; then b + c -d, being squared, produces bb+2bc - 2bd+cc-2cd + dd, which I find to be the fame in all it's Terms with the given Power, although they stand in a different Pofition; confequently b + c - d is the true Root required. Example 2. It is required to extract the Square Root of a+ -2aabb+bt. Here are but two fingle Powers, viz. a and bt, whose Square Roots are a a, and bb. And becaufe in the given Power there is - 2 a abb, therefore I conclude it must either be aa-bborbb-aa. Both which, being involved, will produce a+-2aabb+b+; confequently the Root fought may either be aa-bb, or bb-a a, according to, the Nature or Design of the Question from whence the given Power was produced. Example 3. Let it be required to extract the Square Root of 36 aaaa + 108 a a +81. Here the two fingle Powers are 36 aaaa, and 81, whose Roots are 6 a a and 9. And because the Signs are all + therefore I suppose the Root to be 6 aa +9, the which being involved doth produce 36 a+ + 108 aa+81; conlequently 6 a a +9 is the true Root required. Example 4. Suppose it were required to extract the Cube Root of 125 aaa + 300 a ae - 450 a a + 250 aee - 720 a e +64eee+540a -288ee432e-216. In this Example there are three distinct Powers, viz. 125 a a a, 64 eee, and -216. And the Cube Root of 125 aaa is 5a; of 64eee is 4e; of -216 is- 6. Wherefore I suppose the Root fought to be 5 a +4-6, which being involved to the third Power, does pro duce duce the fame with the given Power; confequently 5 a+48-6 is the Cube Root required. But if the new Power, raised from the supposed Root (being involved to it's due Height) should not prove the fame with the given Power, viz. if it hath either more or fewer Terms in it, &c. then you may conclude the given Power to be a Surd, which muft have it's proper Sign prefixed to it, and cannot be otherwife ex pressed, until it come to be involved in Numbers. Example 5. Suppose it were required to extract the Cube Root of 27 aaa54baa8bbb. Here are two diftinct and perfect Cubes, viz. 27 aaa, and 8bbb, whose Cube Roots are 3 а and 26. Wherefore one may suppose the Root fought to be 3 a +26, which being involved to the third Power, is 27 aaa +54baa+36bba+8bb. Now this new raifed Power hath one Term (viz. 36bba) more in it than the given Power hath; but this being a perfect Cube, one may therefore conclude the given Power is not so, viz. it is a Surd, and hath not fuch a Root as was required, but must be exprefled, or fet down, Thus 27 aaa+54 baa+86bb, If these Examples be well understood, the Learner will find it very easy by this Method of proceeding, to discover the true Root of any given Power whatsoever. CHAP. III. Of Algebraick Fractions, or Broken Duantities. Sect. 1. Rotation of Fractional Quantities. Fractional Quantities are expressed or fet down like Vulgar in common Arithmetick. How they come to be so, see Cafe 4, in the last Chapter of Division. These Fractional Quantities are managed in all re spects like Vulgar Fractions in Common Arithmetick. Y 2 Sect. Sect. 2. To Alter or Change different Fradions into one Denomination, retaining the same Value. RULE. MULTIPLY all the Denominators into each other for a new Denominator, and each Numerator into all the Deno minators but it's own for new Numerators. Let EXAMPLES. a d it be required to bring and into one Denomination. C First a x c, and d x b, will be the Numerators, and b x c will Sect. 3. To Bling whole Duantities into Fractions of a given Denomination. RULE. MULTIPLY the whole Quantities into the given DenoNumerator, which fubfcribe the given Denominator, and you will have the Fraction required. EXAMPLES. Let it be required to bring a + b into a Fraction, whose Denominator is d-a. First a + b xd-ais da +bd-aa-ba: Then dabd-aa-ba d-a is the Fraction required. Again 6+ will be dbta. Anda - a will be aa-da b a d d 2aa Also a+b+a+bb will be 200 a-b When 165 When whole Quantities are to be set down Fraction-wife, ab subscribe an Unit for the Denominator. Thus ab is I And Sect. 4. To Abbreviate, or Reduce Fractional Quantities into their lowest Denomination. RULE. Divide both the Numerator and Denominator by their greatest common Divisor, viz. by fuch Quantities as are found in both; and their Quotients will be the Fraction in it's lowest Term. dc aa is abbb.bb bdc is And a+=a+d. C In such single Fractions as these, the common Divisors (if there be any) are easily discovered by Inspection only; but in compound Fractions it often proves very troublesome, and must be done either by dividing the Numerator by the Denominator, until nothing remains, when that can be done : or else finding their common Meafure, by dividing the Denominator by the Numerator, and the Numerator by the Remainder, and so on, as in Vulgar Fractions (Sect. 4. Page 51.) Suppose EXAMPLES. Then cd-dd) aac-aad aac-aad 0 In this Example it so happens that the Numerator is divided just off by the Denominator; but in the next it is other wife, and requires a double Division to find out the common Measure, viz. Let it be required to reduce aaa-abb aa+2ab+bb First a a+2ab+bb) aaa-abb (a to it's lowest Terms. aaa+2aab+abb -2aab-2abb the Remainder. Then-2aab-2a6b) aa+2ab+bb (-. Hence it appears that - 2 a ab-2abb is the common Mea fure; by which aaa-abb being divided. I 26 2 a I - is the new Denominator. But-27+===20+26 = 2 a -a+b 26 -a-b 2ba 46 -20-26 26 2a 4 ba the Denominator. Let both be multiplied with 2ba, and you will have a a + ab the Numerator. -a-b the Denominator. the Signs of all the Quantities, it will be tion required. That is, aa-ab Again, let it be required to reduce Or changing the new Frac a+b ddd-bbb The common Measure of this Fraction will be the easiest found (as appears from Trials) by dividing the Denominator by the Numerator, &c. Thus, Hence it appears that bd - bb is the common Measure that will divide both the Numerator and the Denominator. Confequently |