Here 81-9, and the square root of a or a= ૭. Ex. 2. What is the square root of a2 −6a/b+9b. a2—6a/b+9b(a—3/b a2 2a-3/b)-6a√b+9b Ex. 3. Find the square root of 93/3. Ans. 3/3. Ex. 4. Find the 4th root of a2. Ex. 5. Find the cube root of (5a2 — 3x2)3. Ans. &/o Ans. (5a3-3x2). Ex. 6. Required the cube root of ja3b. Ans. ab. Ex. 7. What is the fifth root of 323/x5. Ex. 8. What is the 4th root of 16a2x. Ans. 23/x. Ans. 2/ax. Ex. 9. What is the nth root of anx2. 1 2 Ans. amx mn. Ex. 10. It is required to find the cube root of a3 -3a2/x+3ax-xx. Ans. a-√x. IV. METHOD OF REDUCING A FRACTION, WHOSE DENOMINATOR IS A SIMPLE OR A BINOMIAL SURD, TO ANOTHER THAT SHALE HAVE A RATIONAL DENOMINATOR. 350. A fraction, whose denominator is a simple α surd, is of the form; where may represent any Vx 4 rational quantities whatever, either simple or com pound; thus, bc α c-d &c. √ ab' / (a2 —b) ' \/ (a+y) are fractions, whose denominators are simple surd quantities. 351. It is evident that, if a surd of the form be multiplied by/x-1, the product shall be rational; since xXx−1 =" / (x, Xx"−1)="/x"=x ; in like manner, if (a+x) be multiplied by (a+x)2, the product will be a+x. 352. Hence, if the numerator and denominator of a fraction of the form be multiplied by x2-1, the α -1 result will be a fraction, whose denominator shall be α the form be multiplied by /x-1, it becomes Vx quantity. a fraction whose denominator is a rational 353. Compound surd quantities are such as consist of two or more terms, some or all of which are irrational; and if a quantity of this kind consist only of two terms, it is called a binomial surd; and a fraction whose denominator is a binomial surd, is, in general, of the form 3C Va ab 354. If a multiplier be required, that shall render any binomial surd, whether it consist of even or odd roots, rational, it may be found by substituting the given numbers, or letters, of which it is composed, in the places of their equals, in the following general formula: Binomial, Va±b, Multiplier, a1="/a2b+"/an-3b2==~/an1b3+ &c., where the upper sign of the multiplier must be taken with the upper sign of the binomial, and the lower with the lower; and the series continued to n terms. This multiplier is derived from observing the quotient which arises from the actual division of the numerator by the denominator of the following fractions: thus, I. n ·= x2-1+xn--2y+xn-3y2+&c. . +y"-1 to n terms, whether n be even or odd, (Art. 108). xn-yn ww II. · = x2-1 —— x2--2y + x2-3y2 — &c. . . — yn--1. x+y to n terms, when n is an even number, (Art. 109). x+y". III. · = x2-- 1 — x2-2y+x2-sy3 —&c... +ya--1. x+y to n terms, when n is an odd number, (Art.110). 355. Now let x"=a, y"=b; then, (Art. 116), x=/a, y=/b, and these fractions severally a-b become a-b and a+b Va+63 ; and by the application of the rules in the preceding section. we have x-1="/a1--1 ; xn--2="/an--2; xn-s="/an-3, &c. also, y2="/b2; y3="/b3; &c.; hence, x-2y= "/an-2×"/b="/a"-2b; x-3y2="/an-3 × "/b2 = "// -62; &c. By substituting these values of x, x-2y, xn-sy2, &c., in the several quotients, we have a-b :="/a1 ̈1+"/a1-2b+"/an--3b2+&c. ...+. Va-b n b- to n terms; where n may be any whole num a±b Va+b terms b and Wan-sf2 -&c. ... 6-1 ton terms; where the b-- have the sign +, where n is an odd number: and the sign- when n is an even number, 356. Since the divisor multiplied by the quotient gives the dividend, it appears from the foregoing operations that, if a binomial surd of the form "/a/b be multiplied by "/a1+"/an-2b+ &c. • +"/b1 (n being +/bn--1 any whole number whatever), the product will be a-b, a rational quantity; and if a binomial surd of the form a+b be multiplied by an1- ~/a-2b+ //--362 - &c. ... b-1, the product will be a+b or a-b, according as the index n is an odd or an even number. 357. Hence it follows, that, if the numerator and denominator of the fraction (Art. 353), be multiplied by the multiplier, (Art. 354), it becomes another equiva lent fraction, whose denominator shall be rational. There are some instances, in which the reduction may be performed without the formal application of the rule, which will be illustrated in the following examples. Ex. 1. Reduce √20+√12 √5-√3 rational denominator. to a fraction with a To find the multiplier which shall make √5-√3 rational, we have n=2, a=5, b=3; .'. (Art. 354), "/a"-1+"/a"-2b= (since ana22=a°=1) √5+ √20+√12√5+√3_16+4√15 √5-√3 √5+√3 √3; ... 15. = 2 =8+2 358. This multiplier, √5+√3, could be readily ascertained, without the application of the formula. by inspection only; since the sum into the difference of two quantities gives the difference of their squares; also the multiplier that shall render a+brational, is evidently a-b. In like manner, a trinomial surd may also be rendered rational, by changing the sign of one of its terms for a multiplier; and a quadrinomial surd by changing the signs of two of its terms, &c. In the first place, √5+√3-√2 √5+√3+√2 2(√5+√3+√2) √5+√3+√2. -3+ √15 6+2/15 3+15 −3+√15 .(√5+√3+√2)×(−3+√15) is the fraction re To find the multiplier which shall make 3/3—3/2 rational, we have n=3, a=3, b=2; .."Van-1+ ~/n--2b+n/bn-1=3/9+/6+3/4. Now (2/3-2/2)(3/9+/6+4)=a-b=3-2=1; .. the denominator is 1, and the fraction is reduced to 3/9+6+/4. 359. Hence for the sum, or difference, of two cube roots, which is one of the most useful cases, the multiplier will be a trinomial surd consisting of the squares of the two given terms, and their product, with its sign changed. Ex. 4. Reduce 3/15-4/5 rational denominator. |