4, and the third by 3, ical sign to the 6th, gives and to raise the quantities under each rad4th, and 3d powers respectively, which 24 8 24 √5b2/5+b+ √ a2 + b2 = 2 + √ (a2 + 62)3. 24 6 = 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 Whereas, they should have, by the foregoing rule, 6 √√2 = √(2)2 = √4, and √3 = √(3)3 = √√27. Reduce √2, 4, 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 3√ab and 7√ab; as also, 3a2b2, and 9c3 3/b2, 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, 3a2 √√b ± 2c √ √ b = (3a ± 2c) √b. Dissimilar radicals may sometimes be reduced to similar radicals, by the rùles of Arts. 224 and 225. For example, √48ab2+b√75a = 4b √3a + 5b √3a = 9b √3a. 3 3 3 2. 8ab16a - 3⁄43√/b1 +2ab3 = 2a3⁄43√/b+2ab3√b+2a; 3 3. 3√4a2+2√2a = 3√√2a +2√2a = 53⁄4√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. n n Let it be required to multiply a by . and by raising both members to the nth power, (√ a)" × ('√b)" = ab = P" ; and by extracting the nth root, √ a × √√b = P = √ab; that is, the product of the nth roots of two quantities, is equal to the nth root of their product. n Let it be required to divide a by b. If we designate the quotient by Q, we have and by raising both members to the nth power, that is, the quotient of the nth roots of two quantities, rs 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. 3a8a2 × 2b√4a2c = 6ab 32a+c = 12a2b√2c. 3ab5b2c = 15ab × 24/8b+c3. 5. Multiply √√√3 by I 6 Multiply 2√15 by 3/10. 3 X Ans. 8 3 10. Multiply √2, 3, and 5, together. 3 Ans. 12/648000. 11. Multiply √ √ and 14/6, together. 7 12. Multiply (1√√+5√) by (√ √ +2√√). m 229. By raising a to the nth power, we have m ("Va)"="√a × Vā× √a... = √a, by the rule just given for the multiplication of radicals. Hence, for raising a radical to any power, we have the following RULE. Raise the quantity under the sign to the given power, and affect the result with the radical sign, having the primitive index. If it has a co-efficient, first raise it to the given power. 2. (3/2a)5=35. √√(2a) = 243 √32a5 = 486a 3/4a2. : When the index of the radical is a multiple of the power to which it is to be raised, the result can be simplified. = For, √√2a√√√2a (Art. 214): hence, in order to square √2a, we have only to omit the first radical, which gives (√2a)2 = √2a. Again, to square √36, we have √36√√√36: hence, = Consequently, when the index of the radical is divisible by the exponent of the power to which it is to be raised, perform the division, leaving the quantity under the radical sign unchanged. Let it be required to extract the mth root of the radical Va. We have (Art. 214), Hence, to extract the root of a radical, multiply the index of the radical by the index of the root to be extracted, leaving the quantity under the sign unchanged. This rule is nothing more than the principle of Art. 214, enunciated in an inverse order. When the quantity under the radical is a perfect power, of the degree of either of the roots to be extracted, the result can be simplified. |