Whence it follows that in extracting the Roots of all compound Quantities, there must be confidered, 1. How many different Letters (or Quantities) there are in the given Power. 2. Whether the fingle Powers of each of thofe 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 thofe fingle Roots together with the Sign †, and involve them to the fame Height with the given Power; that being done, compare the new raifed Power with the given Power; and if they are alike in all their refpective 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 2 c d + b b −2bddd. 2cd+bb In this Compound Square, there are three diftinct Powers, viz. b b, c c, d d, whole fingle Roots are b, c, d, wherefore I fuppofe the Root, fought to be b+c+d, or rather bed, because in the given Power there is 2 c d, and 2 bd, therefore I conclude it is d; then b+c-d, being fquared, produces b b + 2 be bb2bc -2bd+cc2cddd, which I find to be the fame in all it's Terms with the given Power, although they ftand 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+ — 2a a b b + b+. Here are but two fingle Powers, viz. a and bt, whofe Square Roots are a a, and b b. And because in the given Power there is 2. a abb, therefore I conclude it must either be a abborbb-a a. Both which, being involved, will produce a+ — 2 aabb +b+; confequently the Root fought may either be аа- bb, or b b a a, according to, the Nature or Design of the Queftion 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, whofe Roots are 6 a a and 9. And because the Signs are all therefore I fuppofe the Root to be 6 a a +9, the which being involved doth produce 36 a++ 108 aa +81; conLequently 6 a a +9 is the true Root required. Example 4. Suppofe it were required to extract the Cube Root of 125 a a a + 300 a a e 450 aa 250 a e e- 720ae +64eee+ 540 a 288 c e +432 e 216. In this Example there are three diftinct Powers, viz. 125 a a a, 64 e e e, and -216. And the Cube Root of 125 a a a is 5a; of 64 eee is 4e; of 216 is 6. Wherefore I fuppofe 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+4e-6 is the Cube Root required. But if the new Power, raifed from the fuppofed Root (being involved to it's due Height) thould 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 expreffed, until it come to be involved in Numbers. Example 5. Suppofe it were required to extract the Cube Root of 27 aaa+54baa +8bbb. Here are two diftinct and perfect Cubes, viz. 27 a a a, and 8bbb, whofe Cube Roots are 3 a and 2 b. Wherefore one may fuppofe the Root fought to be 3 a 26, which being involved to the third Power, is 27 aa a +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 fo, viz. it is a Surd, and hath not fuch a Root as was required, but muft be exprefled, or fet down, If thefe Examples be well underflood, the Learner will find it very eafy by this Method of proceeding, to difcover the true Root of any given Power whatfoever. CHA P. III. Of Algebraick Fractions, or Broken Duantitics. Sect. 1. Notation of Fractional Quantities. FRational Quantities are expreffed or fet down like Vulgar Fractions in common Arithmetick. a 2 b c 5b-4a Numerators. Thus {, 4d+7b Denominators. d' 4d +7b How they come to be fo, fee Cafe 4, in the laft Chapter of Divifion. These Fractional Quantities are managed in all re spects like Vulgar Fractions in Common Arithmetick. Y 2 Se&t. Sect. 2. To Alter or Change different Fractions into one Denomination, retaining the fame 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. EXAMPLES. Firft a xc, and d x b, will be the Numerators, and bx c will Sect. 3. To Bling whole Duantities into Fractions of a given Denomination. MUL RULE. ULTIPLY the whole Quantities into the given Denonominator for a Numerator, under which fubfcribe the given Denominator, and you will have the Fraction required. EXAMPLES. Let it be required to bring a + b into a Fraction, whofe Denominator is da. Firft abxda is dab d-aa-ba: When whole Quantities are to be fet down Fraction-wife, a b subscribe an Unit for the Denominator. Thus ab is. And aa-bb, is Sect. 4. To Abbreviate, or Reduce Fractional Quantities into their lowest Denomination. RULE. Divide both the Numerator and Denominator by their greatest common Divifor, viz. by fuch Quantities as are found in both; and their Quotients will be the Fraction in it's lowest Term. Thus aac a a abb b b b And a + с b de =a+d. In fuch fingle Fractions as these, the common Divifors (if there be any) are easily discovered by Infpection only; but in compound Fractions it often proves very troublefome, and must be done either by dividing the Numerator by the Denominator, until nothing remains, when that can be done: or elfe finding their common Meafure, by dividing the Denominator by the Numerator, and the Numerator by the Remainder, and fo on, as in Vulgar Fractions (Sect. 4. Page 51.) EXAMPLES. O In this Example it so happens that the Numerator is divided juft off by the Denominator; but in the next it is other wife, and requires a double Divifion to find out the common Measure, viz. aaa-a - abb to it's lowest Terms. Let it be required to reduceaa+2ab+bb First aa+2ab+bb) a aa—abb (a aaa+2aabtabb -2aab-2abb the Remainder. Then-2aab—2abb) aa+2ab+bb aa+ab ab+bb Hence it appears that 2 a ab. 2 a b b is the common Mea fure; by which aaa-abb being divided. -a-b 26a the Denominator. Let both be multiplied with 2ba, and you will have 1a1 b the Denominator. the Signs of all the Quantities, it will be Or changing aa -ab the new Frac a+b aaab the Numerator. The common Measure of this Fraction will be the easiest found (as appears from Trials) by dividing the Denominator by the Numerator, &c. Thus, ddbb) ddd-bbb (d ddd-bbd 1 + b b d — b b b ) d d — b b d Hence it appears that b d bb is the common Measure that will divide both the Numerator and the Denominator. Confequently |