CASE III. To involve or raise Surd Quantities to any power. 284. Involve the rational part into the proposed power, then multiply the fractional exponents of the surd part by the index of that power, and annex it to the power of the rational part, and the result will be the power required. Compound surds are involved as integers, observing the rule of multiplication of simple radical quantities. Ex. 1. What is the square of 2√✓a? The square of 21/a=(2a1)2=2a×a113 —4a. Ex. 2. What is the cube of 3/(a2—b2+√3) ? 1.3 The cube of (a2—b2+√/3)=(a2—b2+√3)3°3 =a2—b2+ √3. 285. Cor. Hence, if quantities are to be involved to a power denoted by the index of the surd root, the power required is formed by taking away the radical sign, as has been already observed. Ex. 3. What is the cube of 2αx. 3 Here (+)2=‡, and (√/2ax)3=(2ax)13 = (2ax)ŝ 1 =(2ax) ×(2ax)2; ::×2ax ×(2ax)= Ex. 4. It is required to find the square of √a—√b. Na-Nb Navb a-Nab The square a-2ab+b. Ex. 5. It is required to find the square of 33/3. Ans. a3-3a2b+3ab—b、/b. Ex. 10. Required the square of 3+√/5. Ans. 14+65. Ex. 11. Required the cube of/(√/a—√bc). CASE IV. Ans. bc-a. To evolve or extract the Roots of Surd Quantities. RULE. 286. Divide the index of the irrational part by the index of the root to be extracted; then annex the result to the proper root of the rational part, and they will give the root required. If it be a compound surd quantity, its root, if it admits of any, may be found, as in Evolution. And if no such root can be found, prefix the radical sign, which indicates the root to be extracted. Ex. 1. What is the square root of 81√α. Here 819, and the square root of a or a =a*÷2= a*x*=a*=\/a; .•.~^/(81^/a)=9^/a, or 9a1. Ex. 2. What is the square root of a2-6a/b+9b. Ans. amxm. Ex. 10. It is required to find the cube root of a3-3a3√x+ 3ax-x/x. Ans. a-x. § IV. METHOD OF REDUCING A FRACTION, WHOSE DENOMI NATOR IS A SIMPLE OR A BINOMIAL SURD, TO ANOTHER THAT SHALL HAVE A RATIONAL DENOMINATOR. 287. A fraction, whose denominator is a simple surd, is of √ab'n/(a2—b)'3/(a+y) &c. are fractions, whose denominators are simple surd quantities. 288. It is evident that, if a surd of the form / be multiplied by -1, the product shall be rational; since /x/(xxx-1)="/x"=x; in like manner, if (a+x) be multiplied by (a+r)2, the product will be a+x. 289. Hence, if the numerator and denominator of a fraction α of the form be multiplied by n/x-1, the result will be a frac tion, whose denominator shall be rational. Thus, let both the numerator and denominator of the fraction α be multiplied by, and it becomes ; and by √x multiplying the numerator and denominator of the fraction 3/16 (a+x)' by (a+x), it becomes = V/[b(a+x)a]_b3(a+x)* (a+x)3 a+x Or, in general, if both the numerator and denominator of a α fraction of the form be multiplied by -1, it becomes an/x-1 a fraction whose denominator is a rational quan 290. 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 Ꮖ surd, is, in general, of the form ab 291. 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, ab. Multiplier, /1/an-2b+/an¬3b2="/an-4b3 + &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. xn-yn -=x1+x-2y+2n-Sy2+&c. .+yn-1 to n terms, whether n be even or odd, (Art. 108). terms, when n is an even number, (Art. 109). terms, when n is an odd number, (Art. 110). 292. Now let xn=a, yn=b; then, (Art. 116), z=*/α, y=/b, and these fractions severally become a-b and a-b /a-/b a+b ; and by the application of the rules Va+b' Va+w/b in the preceding section we have x-1="/an−1; x^2=*/an-9 27-8/a-3, &c. also, y=b2; y3="/b3 ; &c.; hence, xn-2y="/an-2X"/b=x/an-2b; xn-sy2==~/an3×"/b2="/a n-363; &c. By substituting these values of x-1, x-2y, xn-3y3, &c., in the several quotients, we have /an-2b+1/an-sf2+&c. n may n a- -b +/- to n terms; where be any whole number whatever. And /an-1-/an-2b+/an-362-&c. ... a+b n - to n terms; where the terms b and /b-1 have the sign +, when ʼn is an odd number and the sign : -, when n is an even number. 293. 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+ 2/an-2b+&c..+/bn-1 (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 /a”— /a-2b+/-3b2—&c....±7/6-1, the product will be a+b or a-b, according as the index n is an odd or an even number. 294. Hence it follows, that, if the numerator and denominator of the fraction (Art. 290), be multiplied by the multiplier, (Art. 291), it becomes another equivalent 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 nominator. √/20+√/12 to a fraction with a rational de To find the multiplier which shall make we have n=2, a=5, b=3; ... (Art. 291), =(since a”—2=a2-2=a°=1) √5+√3; √5+√3_16+4√15 √5+3 =8+2/15. 2 295. 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 ab. 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. -3+/15 (√5+√3+√2) ×(−3+√ 15) is the fraction required. 6 1 to a fraction with a rational Ex. 3. Reduce 7/3/2 denominator. To find the multiplier which shall make 3/3-3/2 rational, we have n=3, a=3, b=2; .. */an-1+~/n-2b+~/b-1=" 3/9+3/6+/4. Now (2/3-2/2)(†/9+6+/4)=a— b=3—2=1; |