M duce the fame with the given Power; confequently 50+40-6 is the Cube Root required. But if the new Power, raised from the fuppofed Root (being involved to it's due Height) fhould 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 aa, 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+36 b b a +8bb. Now this new raised 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 must be expreffed, or fet down, Thus 27 aaa+54baa +8bbb. If these Examples be well understood, the Learner will find it very eafy by this Method of proceeding, to difcover the true Root of any given Power whatsoever. CHA P. III. Of Algebzaick Fractions, or Broken Duantities, Rational Quantities are expreffed or fet down like Vulgar Fractions in common Arithmetick. Thus { 2 b c 5b-4a Numerators. 4d+7b Denominators. How they come to be fo, fee Cafe 4, in the laft Chapter of Divifion. These Fractional Quantities are managed in all reSpects like Vulgar Fractions in Common Arithmetick. Y 2 Seat. 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 Densminators but it's own for new Numerators. Firft a xc, and d x b, will be the Numerators, and bx c will To Bring whole Duantities into Fractions of a given Denomination. RULE. MULTIPLY the whole Quantities into the given Deno nominator 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 a+bxd—a is dab d—aa-ba: da+bda a ba d-a is the Fraction required. Then a b When whole Quantities are to be fet down Fraction-wife, fubfcribe an Unit for the Denominator. Thus ab is is. And aa-bb, is a a b b &c. I Sect. 4. To Abbzeviate, or Reduce Fractional Quantities into their loweft Denomination. Dlv RULE. Ivide both the Numerator and Denominator by their greatest common Divifer, viz. by fuch Quantities as are found in both and their Quotients will be the Fraction in it's lowest Term. abb b b b is is dc d' a b c In fuch fingle Fractions as thefe, the common Divifors (if there be any) are easily discovered by Infpection 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 elfe finding their common Measure, by dividing the Denominator by the Numerator, and the Numerator by the Remainder, and fo on, as in Vulgar Fractions (Sect. 4. Page 51.) In this Example it fo happens that the Numerator is divided juft off by the Denominator; but in the next it is otherwife, and requires a double Divifion to find out the common Measure, viz. Let it be required to reduce aa+2ab+66 2 a b b is the common Mea Hence it appears that — 2 a a b. fure; by which a aa-abb being divided. .b 2ba the Denominator. Let both be multiplied with 26a, The common Measure of this Fraction will be the eafieft found (as appears from Trials) by dividing the Denominator by the Numerator, &c. Thus, Hence it appears that bdbb is the common Measure that will divide both the Numerator and the Denominator. Confequently Consequently b d →→bb) ddib+1, is the new Numerator. +db-bb And bd-bb) ddd-bbb (dd ddd-ddb+d+b the new Denominator. Let both be multiplied with b, and then you will have d+b the Numerator, dd+bd+bb the Denominator, } of the Fraction required. But if after all Means used (as above) there cannot be found one common Measure to both the Numerator and Denominator; then is that Fraction in it's leaft Terms already. Note, Thefe Operations will be understood by a Learner after he hath paffed thro' Multiplication, and Divifion of Fractions. Sect. 5. Addition and Subtraction of Fractional Duantities. THE HE given Fractions being of one Denomination, or if they are not, make them fo, per Sect. 4. Then, RULE. Add or fubtract their Numerators, as Occafion requires, and to their Sum or Difference, fubfcribe the common Denominators as in Vulgar Fractions. |