rations in article 64. He should be informed, however, that here numerical surds are the things chiefly treated of, and that for the reduction and simplification of these the former article makes no provision. His attention should also be directed to the fact, that, in the examples of division here proposed, the divisor is always supposed to be a monomial surd; when it is binomial or trinomial, &c. the simplification is not so easily effected: to obtain the quotient in such a case, in the least complicated form, requires distinct rules. In practice, the surd divisors are seldom more than binomial; and for this case we shall now give the method for obtaining the quotient in the desired form, and shall afterwards say a word or two on the treatment of trinomial surd divisors. 71. ON DIVISION BY BINOMIAL SURDS. We have already seen that, to divide by a monomial surd, the first step, after having brought dividend and divisor under a common index, is to render the divisor rational as respects that index; so that the root indicated may be actually extracted, and the divisor thus freed from the radical sign; this is effected by the introduction of a suitable rationalising factor, as already exemplified. When the divisor is binomial, the object is the same; we must seek a multiplier that will render that divisor a rational quantity, or free it from surds; the present article, therefore, has for its principal object the determination of multipliers that will make binomial surds rational. Such binomials usually consist of either the sum or difference of two square roots, or of the sum or difference of two cube roots; that is, they are usually of one or other of the forms √a±√b, a±3/b. the former case the rationalising multiplier is itself a binomial surd, suggested by the property that the product of the sum and difference of any two quantities is the difference of the squares of those quantities; so that if a+b be multiplied by va√b, or √a-√b by √ a+√b, we know by this property that the product will in either case be a-b, a rational quantity. When the binomial surd consists of two cube roots, the rationalising multiplier will be trinomial; it will consist of the squares of both the proposed surd terms and of their product with changed sign; this is confirmed by actual multiplication, for (Va±vb)(V/a2=Vab+$/b2)=a±b. In This general formula, together with the corresponding one for binomial quadratic surds, as surds affected with the square root are called, which formula, as just noticed, is (√ a±√b) (√ a√b)=a—b, convey, in a symbolical form, the rules to be used in all particular cases of the kind here considered, and render any formal directions in words unnecessary; we shall therefore at once proceed to examples. (1) It is required to divide 3 by 27-35. 3(2√7+3√5) 3(2/7+3√5) = 3 2√7—3√5 ̄ ̄ ̄ ̄(2√7—3√5)(2√7+3√5) 28-45 3 (2√/7+3√/5). 17 5 (2) It is required to convert into a fraction with a rational denominator. 5 37-3/5 5($49+/35+25) __ 5(†49+✔35+325) 37-35 (4) Required a multiplier that will render 3+4 rational. (V3+V4)(√3—√4)=√3—√4 and (√3—√4)(√3+√4)= 3-4=-1; hence the sought multiplier consists of the two factors 72. When the divisor is a trinomial surd of the form √a+ b+c, that is, a trinomial quadratic surd, it may be rationalised after the manner following: 1st (Va+b+c) (√ a + √ b−√ c) = (√ a + √ b)2 —c= a+b—c+2√ ab. 21 (a+b-c+2 ab) (a+b—c—2√ ab)=(a+b—c)2-4ab. But it will not be necessary to encumber the page with applications of this process to particular examples, since the operations are only a repetition of those for binomial quadratic surds. A single illustration will no doubt be sufficient. Let it be required to render √5-√ 3+√7 rational. (√5—√3+ √7) (√ 5 —√ 3—√7) = ( √ 5 — √ 3) 2—7= 1-2√ 15; and (1—2√ 15)(1 + 2 √ 15)=1—4 × 15—=—59. Hence the rationalising multiplier is (√5-√3-√7)(1+2√15). If we had taken for the first factor √3+√5+√7, the second would have been 9—2√35, and the product of the two would have been the same as before. For the means of rationalising binomial surds, the two terms of which are different roots, the learner is referred to the larger treatise on algebra; but there are two or three general theorems in reference to quadratic surds, which must not be passed over here; they are both interesting and useful. 1. The square root of a quantity cannot be partly rational and partly a quadratic surd. For if we could have such a condition as a=b+√c, √ a and c being surds, then, by squaring each side, we should have a=b2+2b√c+c; and consequently, √c= a-b2-c ; that is, It a surd equal to a rational quantity, which is impossible. follows from this, that in approximating to the square roots of surd numbers, it can never happen that the interminable decimals omitted can, in any two cases, be the same; for if they could, the difference of the two roots would be finite and rational, and, consequently, one of the surds would be equal to a rational number plus the other surd, which is here shown to be impossible. 2. In every equation of the form a+√b=x+√y, where a and a are rational quantities, and ✅b and ✅y surds, a must be equal to x, and therefore b to y; for if a and x had any difference, then, by transposing the a or the x, we should have the square root of a quantity equal to a quantity partly rational and partly a quadratic surd, which we have seen to be impossible. 3. If the square root of a+b, a quantity consisting of two terms, the one rational and the other a quadratic surd, be equal to x+y, where x and y are one or both quadratic surds; that is, if √(a+√b)=x+y, then will ✔(a-√b)=x—y. For by squaring, the first equation is a+b= x2+2xy + y2, where ay is necessarily a rational quantity, and consequently 2xy must be irrational, otherwise by transposing a, we should have b equal to a rational quantity; hence (by the theorem 2) a= x2+ y2 and b=2xy; consequently, a—√b= x2 +- y2— 2xy=(x—y)2.. √(a—√b)=x—y. 73. To extract the square root of a binomial, one of whose terms is rational and the other a quadratic surd: Let ab be the binomial surd, of which the square root is to be extracted. Put √(a+b)=x+y; and √(a−√ b) — x-y; therefore, by squaring these two equations .. by addition, 2a =2x2+2y2 :. a= x2 + y2......(A). Also, multiplying the same two equations together, √(a+b) (a—b) = x2-y; that is, √ (a2 —b) = x2-y2 ...... (B). .2 From (A) and (B), by addition and subtraction, Consequently, for (a+b) and (a-vb), equal respectively to x+y and x-y, we have the expressions √(a−√b)=√ 2 2 -√ (a2—b) Hence, if the numbers or quantities a, b, involved in the proposed binomial surd, be such as to render the expression a2-b a complete square, as c2, then √(a+√b)=√a‡c+√=; 2 2 2 where the complex surd ✔(ab) is reduced to two simple surds; but if the condition a2-b=a square, do not hold, then nothing is gained in the way of arithmetical simplicity by transforming the expressions on the left into those on the right: the transformation is in fact the exchanging of one complex surd for two; hence, in applying the formula to numerical examples, the first thing to be done is to ascertain whether a2-b is a square; if it be, then calling it c2, we are to use the forms of reduction last given; if it be not, the formulæ are practically useless. (1) Required the square root of 8+39. Here 82-39-64-39-25=52; Hence, by means of a table of square roots, √(8+√39), thus transformed, may be determined with but little trouble; we have only to take from the table 26 and 6, and to find half their To obtain the result from the unreduced form, we should have actually to extract the square root of 8+39 after ✔39 had been got from the table. sum. (2) Required the square root of 4+2√3. Here the square of the first term, minus the square of the second, is 4-2 4o—12—4—2o .. √ (4 + 2√3)=√ 4+2 + √ 4 — 2 = √3+1. 2 2 In this example the required square root is pretty obvious without any reference to a formula; for the proposed expression is at once seen to be the same as 3+2√3+1=(√3+1)o. NOTE. In certain cases wherein the condition a2--b-c2 fails, a simple transformation will enable us to exhibit the root in the form of a binomial surd; for example: |