Ex. 5. Let x2y'a be expressed with a negative exponent. 1 Ans. xya ̈}• CASE III. To involve or raise Surd Quantities to any power. RULE. 347. 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, observ ing the rule of multiplication of simple radical quantities. Ex. 1. What is the square of 2/a? The square of 2a=(2a)2 = 2a × a22=4α. Ex. 2. What is the cube of (a2 —b3+√3) ? The cube of /(a*b2 +√/3)=(a2 —b2 +√/3)3°3 =a2-b2+√3. 348. 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 (Art. 326). Ex. 3. What is the cube of√2ax. Here (1)3, and (√2ax)3 = (2ax)1⁄2·3 #(2ax)ŝ Ex. 4. It is required to find the square of ✔a b. Ex. 5. It is required to find the square of 3/3. Ans. 9/9. Ex. 9. Required the cube of a-√b. Ans. a3-3a2b+3ab-b✔✅b. Ex. 10. Required the square of 3+5. Ans. 14+65. Ex. 11. Required the cube of −3⁄4/(√a−√bc). Ans. bc-α. CASE IV. To evolve or extract the Roots of Surd Quantities. RULE. 349. 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 Évolution. 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/a. Here ✓819, and the square root of a or a2 = 9a*. Ex. 2. What is the square root of a2 -6a/b+9b. a2-6ab9b(a-3/b a2 2a-3b)-6a/b+9b Ex. 3. Find the square root of 93/3. Ex. 4. Find the 4th root of fa2. Ans. 33. Ex. 5. Find the cube root of (5a2 — 3x2)3. Ans. Ans. (5a3-3x2). Ex. 6. Required the cube root of a3b. Ans. tab. Ex. 7. What is the fifth root of 323/x5. Ex. 8. What is the 4th root of 16a2x. Ans. 23/x. Ans. 2/a1x. Ex. 9. What is the nth root of */ax2. 1 2 Ans. amx mm Ex. 10. It is required to find the cube root of a -3a2x+3ax-xx. 3 Ans. a-√x. § IV. METHOD OF REDUCING A FRACTION, WHOSE DENOMINATOR IS A SIMPLE OR A BINOMIAL SURD, TO ANOTHER THAT SHALL HAVE A RATIONAL DENOMINATOR, 350. A fraction, whose denominator is a simple surd, is of the form n α ; where may represent any rational quantities whatever, either simple or compound; thus, C d √ab' / (a2 —b) ' \/\/ (a+y) &c. are fractions, whose denominators are simple surd quantities. 351. It is evident that, if a surd of the form n-1. be multiplied by "/"-1, the product shall be rational; since xXx11 =1 / (x X x1−1)="/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 be multiplied byx", the a fraction of the form n α result will be a fraction, whose denominator shall be rational. Thus, let both the numerator and denominator of α fraction be multiplied by √x, and it becomes √30 α the form be multiplied by ", it becomes V/x a2x2-1 quantity. 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 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, /^/b, 3 Multiplier,a="/an-2b+"/an¬3b2="/a113+ &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. -= x2--1+xn--2y+x2-Sy2+&c. • +y^-1 to n x-y terms, whether n be even or odd, (Art. 103). II. · = x2— 1 — x2--2 y + x2-3y2 — &c. x+y to n terms, when n is an even number, (Art. 109). 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 ab a+b the application of the rules in the preceding section. we have x-1="/an-1; xn--2="/a′′-2; x2-s="/an-3,&c. also, y2="/b2; y3="/63; &c.; hence, x-2y= "~//an-2×"/b="/a1--2b; xn--3y2="/a1-3 × "/b2 =^/ a-b; &c. By substituting these values of x-1, x-2y, xn--3y2, &c., in the several quotients, we have a- -b n +&c• ="/a"--"+"/a"-2b+"//an3b2+&c. ...+ b" to n terms; where n may be any whole num |