2d. Extract the square root of the perfect square, and prefix it as a coefficient to the other part placed under the radical sign. To determine whether any numeral contains a factor that is a perfect square, divide it by either of the squares 4, 9, 16, etc. Reduce to their simplest forms the radicals in each of the following examples: 1. √12, √18, √45, √/32, √50a3, √72a2b3. Ans. 2√3, 3√2, 3√5, 4√2, 5α√/2a, bab√2b. 2. √245, 448, 810, 50763c2, √/1805a+b2. Ans. 7√/5, 8√7, 9√/10, 13bc√/36, 19a2b√5. In a similar manner, polynomials may sometimes be simplified. (3a3-6a2c+3ac2)=√3a (a2-2ac+c2)=(a–c)√3a. Thus, 3. √(a3—a2b), √ax2—6ax+9a, √(x2-y2)(x+y). Ans. aya-b), (x−3)√ā, (x+y)√(x—Y). To reduce a fractional radical to its simplest form, 1st. Render the denominator of the fraction a perfect square by multiplying or dividing both terms by the same quantity. 2d. Separate into two factors, one of which is a perfect square. 3d. Extract the square root of this factor, and write it as a coëfficient to the other factor placed under, the radical sign. 200. To reduce radicals of any degree to the most simple form. The principle of Art. 199 is, evidently, applicable to radicals of any degree. Thus, 1. Reduce 54 to its most simple form. 7/54=7/27/2=27×72=3p2. 3 Similarly, = ?×?×?=&!;=✔ 2×18={√ 18. Reduce each of the following to its simplest form: 2. 40, 81, 128ac3, 162m1n3, † 144. Ans. 25, 3c3c, 4a2cp/2c2, 3mn † 6mn2, 2†/9. Ans. 174, 176, 1736, 1715, 11/54, 11/768, 11/32. 201. The mnth root of any quantity may be simplified when it is a complete power of the mth or nth degree, as shown, (Art. 192.) Thus, 9a29a2√3a. Also, √/a2—2ab+b2=√√ √ a2_2ab+b2—ya—b. Reduce each of the following to its simplest form 1. /36a2c2, 81m2n, /4a2, 16a2c4, 12563. Ans. bac, 3nm, 3/2α, ac2, √5b. Case II.-TO REDUCE A RATIONAL QUANTITY TO THE FORM OF A RADICAL. 202. If we square a, and then extract the of the square, the result is evidently a. square root That is, a=√a2=a2. In like manner, a=ā3—a3, and gen Rule for reducing a Rational Quantity to the form of a Radical. Raise the quantity to a power corresponding to the given root, and write it under the radical sign. 1. Reduce 6 to the form of the square root. Ans. Ꮴ 36. 2. 3. -2 to the form of the cube root. Ans. -8. 3ax to the form of the square root. Ans. 9a2x2. 4. m-n to the form of the square root. Ans. /m2-2mn+n2. Similarly, a coëfficient may be passed under the radical sign. Generally, a/b/a""/b="/amb. 5. Express 57, and a2, entirely under the radical sign. Ans. 175, and ab. Va1b. 6. Pass the coëfficient of the quantity 25, under the radical sign. Ans. 40. Case III. TO REDUCE RADICALS HAVING DIFFERENT INDICES TO EQUIVALENT RADICALS HAVING A COMMON INDEX. 203. This is done by multiplying both terms of the fractional exponent by the same number, which, evidently, does not change its value. (Art. 118.) (2a)} Let it be required to reduce 2a, and 36, or and (36) to quantities of equal value, having the same index. √2a=(2a)}=(2a)131⁄2_¥/(2a)1—¥/16a1. Rule.-Reduce the fractional exponents to a common denominator; then the numerator of each fraction will represent the power to which the corresponding quantity is to be raised, and the common denominator the index of the root to be extracted. 1. Reduce 1/3 and 1/2, or 3 and 23 to a common index. 5. √3, a2, and a3. Ans. a1s, 12/a3, and a3. 6. Reduce 33, 24, and 5 to a common index. 9 Ans. 31, 214, 51, or 1/6561, 512, †15625. ADDITION AND SUBTRACTION OF RADICALS. 204. Required to find the sum of 3ƒ1⁄4ã and 5pā. α It is evident that 3 times and 5 times any quantity, must make 8 times that quantity; therefore, 3а÷÷57 ã÷87a. But, if it were required to find the sum of 3/α and 5pā, since Va and a are different quantities, we can only indicate their addition; thus, 3ya+57a. Similarly, 3/2+7√2-4√/2-6√/2. Radicals that are not similar, may often be made so; thus, √12 and v 27 are equal to 21/3 and 31/3, and their sum is 5√3. The same principles apply to the subtraction of radicals. From the above we derive the following Rule for the Addition of Radicals.-1st. Reduce the radicals to their simplest forms, and, if necessary, to a common index. 2d. If the radicals are similar, find the sum of their coefficients, and prefix it to the common radical; but if they are not similar, connect them by their proper signs. Rule for the Subtraction of Radicals.-Change the sign of the subtrahend, and proceed as in addition of radicals. 1. Find the sum of √/448 and √/112. √/448=√/64×7= 112=√16×7= 4√7 By addition, 127, Ans. |