By the definition of a root (Art. 213), we have and by the rule for the raising of powers, (√ax Vox Ve.....)" = ('√√ a)" × ("√ b)" × ('√ c)"... and since the nth powers are equal, the quantities themselves are equal: hence, that is, the nth root of the product of any number of factors, is equal to the product of their nth roots. 1. Let us apply the above principle in reducing to its sim plest form the imperfect power, 54a4b3c2. We have 3 3 54a3b3c2 = 3√27a3b3 × √2ac2 = 3ab2ac2. 2. In like manner, 3/8a2 = 2 3. Also, 6 3 2√2; and 48a5b8c62ab2c √3ac2; = 2√a2 6 192a7bc12=64ac12 x3ab2ac2 3ab. In the expressions, 3ab3/2ac2, 23a2, 2ab2c3ac2, each quantity placed before the radical, is called a co-efficient of the radical. 225. The rule of Art. 214 gives rise to another kind of simplification. 6 Take, for example, the radical expression, 4a2; from this rule, we have and as the quantity affected with the radical of the second degree,, is a perfect square, its root can be extracted: hence, that is, when the index of a radical is a multiple of any number n, and the quantity under the radical sign is an exact nth power, we can, without changing the value of the radical, divide its index by n, and extract the nth root of the quantity under the sign. This proposition is the inverse of another, not less important; viz., the index of a radical may be multiplied by any number, pro-· vided we raise the quantity under the sign to a power of which this number is the exponent. For, since a is the same thing as m √a^, we have, 226. This last principle serves to reduce two or more radicals to a common index. For example, let it be required to reduce the two radicals By multiplying the index of the first by 4, the index of the second, and raising the quantity 2a to the fourth power; then multiplying the index of the second by 3, the index of the first, and cubing a + b, the value of neither radical will be changed, and the expressions will become 3√2a = √2+a+ 12 12 = 12 16a; and (a + b) = 12 √ (a + b)3. Hence, to reduce radicals to a common index, we have the following RULE. Multiply the index of each radical by the product of the indices of all the other radicals, and raise the quantity under each radical sign to a power denoted by this product. This rule, which is analogous to that given for the reduction of fractions to a common denominator, is susceptible of similar modifications. Since 24 is the least common multiple of the indices 4, 6, and 8, it is only necessary to multiply the first by 6, the second by 4, and the third by 3, ical sign to the 6th, gives 24 6 CALCULUS OF RADICALS. 225 and to raise the quantities under each rad4th, and 3d powers respectively, which 24 8 24 √ a = 2√a; √5b = 21/51b*, 3√ a2 + b2 = 21√ (a2 + b2)3. 3 In applying the above rules to numerical examples, beginners very often make mistakes similar to the following: viz., in reducing the radicals 2 and 3 to a common index, after having multiplied the index of the first, by that of the second, and the index of the second by that of the first, then, instead of multiplying the exponent of the quantity under the first sign by 2, and the exponent of that under the second by 3, they often multiply the quantity under the first sign by 2, and the quantity under the second by 3. Thus, they would have 3 6 6 6 6 √2 = √√2×2=√√4, and √3 =√3 x 3 = √9. Whereas, they should have, by the foregoing rule, 3 √√2 = √(2)2 = √√4, and √3 =√(3)3 = √√27. Reduce √2, 4, 1, to the same index. Addition and Subtraction of Radicals. 227. Two radicals are similar, when they have the same index, and the same quantity under the sign. Thus, 3√ab and 7 √ab; as also, 3a23⁄4√62, are similar radicals. In order to add or subtract similar radicals, add or subtract their co-efficients, and to the sum or difference annex the common radical. Thus, 3√6 +23√6=53; also, 3-2 √6 = 3√6. 3a2√√√b± 2c√√b = (3a ± 2c) √ b. Dissimilar radicals may sometimes be reduced to similar radicals, by the rules of Arts. 224 and 225. For example, 1 √48ab2 + b 750 + b√75a=4b √√3a + 5b √3a9b √3a [15] 3 3 2. √8a3b +16a+ — √√/ba + 2ab3 = 2a 3√/b + 2a − b 33⁄4√/b + 2a; b4 = (2a - b)/b+2a. 3. 3√4a2+2√2a = 3 3√√2a + 2 3⁄43√2a = 53√2a. When the radicals are dissimilar and irreducible, they can only be added or subtracted, by means of the signs + or Multiplication and Division. 228. We will suppose that the radicals have been reduced to a common index. and by raising both members to the nth power, (√ a)" × (√b)" = ab = Pn ; and by extracting the nth root, √ax√√b=P="√ab; α that is, the product of the nth roots of two quantities, is equal to the nt root of their product. Let it be required to divide a by T. If we designate the quotient by Q, we have Va and by raising both members to the nth power, that is, the quotient of the nth roots of two quantities, is equal to the nth root of their quotient. Therefore, for the multiplication and division of radicals, we have the following RULE. I. Reduce the radicals to a common index. II. If the radicals have co-efficients, first multiply or divide them separately. III. Multiply or divide the quantities under the radical sign by each other, and prefix to the product or quotient, the common radical sign. 3a√ 8a2 × 2b√4a2c = = 6ab√32a+c = 12a2b√2c. 3 5. Multiply√√ by√ VE 6 Multiply 2√15 by 3/10. Ans. 128. |