Ex. 18. Divide (a+b3)1⁄2 by (a^+b2) §. Ans. (a+b3): Ex. 19. Multiply 4+22 by 2-√2. Ex. 20. Multiply ✔(a−√(b−√3)) by √(a+ √(b−√3)). Ans. (a-b+√3). Ex. 21. Divide a3b-ab2c by a2+a/bc. Ans. ab-bbc. Ex. 22. Divide a1+x by a2+ax√2+x2. Ans. a3-ax/2+x2: 346. It is proper to observe, since the powers and roots of quantities may be expressed by negative exponents (Arts. 86, 311), that any quantity may be removed from the denominator of a fraction into the numerator; and the contrary, by changing the sign of its index or exponent; which transformation is of frequent occurrence in several analytical calculations. 1 Ex. 1. Thus, (since—3—6—3), 65 may be expressed by a2b-3; and (since a2=a2), we have b3b3a2. 63 Ex. 2. The quantity may be expressed by a2 b3c-4e-5. Ex. 5. Let x2y at 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 2/a=(2a)2 = 2a × a3·3=4a. Ex. 2. What is the cube of (a2 —b2 +√3) ? The cube of (a2 —b2+√/3)=(a2 —b3+√3)3°3 =a3-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 of2ax. Here (1)3, and (√2ax)3=(2ax)1⁄2·3 +(2ax)ŝ Ex. 4. It is required to find the square of am b. Ex. 5. It is required to find the square of 33/3. Ex. 9. Required the cube of a-b. Ans. a3-3a2b+3ab-b√√✅b. Ex. 10. Required the square of 3+√5. Ans. 14+6/5. Ex. 11. Required the cube of /(√a−√bc). CASE IV. Ans. bc-a. 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 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 31/a. Here/81-9, and the square root of a or a va 9a =α = /α; •• v/(811/a)=9%/a, or Ex. 2. What is the square root of a2 -6a/4+9b. a2-6a/b+9b(a-3/b a2 2a-3/b)-6a/b+9b Ex. 3. Find the square root of 93/3. Ex. 4. Find the 4th root of a2. Ex. 5. Find the cube root of (5a2-3x2) Ans. 3/3, Ans. a. -3x2) Ans. (5a2-3x2). Ex. 6. Required the cube root of a b. Ans. u/b. Ex. 7. What is the fifth root of 32/5. Ex. s. What is the 4th root of 16a2x. Ans. 23/ Ans. 2/ax. Ex. 9. What is the nth root of "a"x". 1 2 Ans. ac mn Ex. 10. It is required to find the cube root of a -3a2x+3αx-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 x may represent any Vx rational quantities whatever, either simple or compound; thus, 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 "/", the product shall be rational; since X-1 =" / (x × x”—1)="/x"=x; in like manner, if (a+x) be multiplied by (a+x), 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 a result will be a fraction, whose denominator shall be 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 |