n--1 - to n terms; where the terms b and /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/an-1+"/an-2b+ &c. . +/b (n being 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 an- - /an-2b+ -362-&c. ...", the product will be a+b or a―b, according as the index n is an odd or an even number. n 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. 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), ɣn--1+^/an-2b= (since an-2=a2-2=ao= `1) √5+. √20+√12✓5+√3_16+4/15. √5-√3 √5+√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, ; .'. 2(√5+√3+√2) √5+√3+√2-3+ √15 3+15 -3+/15 (√5+√3+√2)×(−3+✓15) is the fraction re 6 6+2/15 To find the multiplier which shall make 3-3/2 rational, we have n=3, a=3, b=2; ../an-it ~/n--2b+n/b?-1=3/9+/6+3/4. Now (1/3-2/2)(3/9+3/6+3/4)=a−b=3—2=1; ... the denominator is 1, and the fraction is reduced to 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. to a fraction with a √15+√5 Ex. 5. Reduce to a fraction with a ra tional denominator. Ans. 3√5+3√x 5-x If √ is irrational, (that is, not a square), the ad - dition or subtraction can be only made by connecting the surds by the signs + or -1, as they are. STURMIUS, in his Mathesis Enucleata, has also given a method similar to the above.. Ex. 4. To transform √2+√3 to a general surd. Ex. 5. To transforma-2x to a universal surd. Ans. (a+4x-4√√/ax). Ex. 6. To transform 3/4+/72 to a universal surd. Ans.3/9. " V. METHOD OF EXTRACTING THE SQUARE ROOT OF BINOMIAL SURDS. 363. The square root of a quantity cannot be partly rational and partly a quadratic surd. If possible, let √n=a+√m; then by squaring both sides, n=a2 + 2a/m+m, and 2a/mn-a2-m; therefore, ✔m= n—a2. m , a rational quantity, which is contrary to 2a the supposition. A quantity of the form a is called a quadratic surd 364. In any equation x+y=a+b, consisting of rational quantities and quadratic surds, the rational parts on each side are equal, and also the irrational parts. If a be not equal to a, let x=a+m; then a+m+ ✔y=a+b, or m+y=√b; that is, b is partly rational, and partly a quadratic surd, which is impossible, (Art. 363); .. x=a, and, y=√b. 365. If two quadratic surds x and y, cannot be reduced to others which have the same irrational part, their product is irrational. If possible, let xy=rx, where r is a whole number or a fraction, Then xy=r2x2, and y=r2x; &• ✔y b'"m ../a+V/6= bm) the nth root = =amm+na'"--1 ×mb'"+&c. "(a'"m+nma'n-1 b'"+&c. · of a rational quantity. Hence the product of a by ✔b is rational if a and b admit of the same irrational part; also, a X/b, or /ax/b2, is rational, if a and b admit of the same irrational part; and, in general, "a"-1 ×/b, or axb-1, is rational, if a and b admit of the same irrational part. 362. It is proper to observe, that, for the addition or subtraction of two quadratic surds, the following method is given in the BIJA GANITA, or the Algebra of the HINDUS, translated by STRACHEY. Thus, to find the sum or difference of two surds, a and b, for instance. RULE. Call a+b the greater surd; and, if a Xb is rational, (that is, a square), call 2/ab the less surd, the sum will be (a+b+2√ab), (=√a±√6), and the difference (a+b-2✓ab). If axb is irrational, the addition and subtraction are impossible; that is, they can only be indicated. Example. Required the sum and difference of √2 and 8. Here 2+8=10=> surd; 2X8=16, ..164, and 2/16=2x4=8=<surd. Then 10+8=18, and 10—8=2; .'. 18 sum, and √2 = difference. ANOTHER RULE. Divide a by b, and write b in two places. In the first place add 1, and in the second subtract 1; a then we shall have √[(√ +1) 3×b] = √a+ √b, and √(√1⁄2 −1)2 ×b]=√a−√b. |