n-1, multiplied by each of the remaining terms, other terms of the development. Hence, we see, that the rule for the cube root will become the rule for the nth root, by first extracting the nth root of the first term, taking for a divisor n times this root raised to the n - 1 power, and raising the partial roots to the nth power, instead of to the cube. (2a-3x)=16a1— 96a3x+216a2x2-216ax3+81x4 | 4× (2a)3=32a3 We first extract the 4th root of 16a4, which is 2a. We then raise 2a to the third power, and multiply by 4, the index of the root; this gives the divisor 32a3. This divisor is contained in the second term 96a3x, 3x times, which is the second term of the root. Raising the whole root to the 4th power, we find the power equal to the given polynomial. 2. What is the 4th root of the polynomial, 81a8c1 + 16h4d4 — 96a2cb3d3 — 216a6c3bd + 216a+c2b2d2. 224. When the monomial or polynomial whose root is to be extracted, cannot be resolved into as many equal and rational factors as there are units in the index of the root, it is said to be an imperfect power. The root is then indicated by placing the quantity under the radical sign, and writing over it at the left hand, the index of the root. Thus, the fourth root of 3ab2+9ac3, is written √3ab2+9ac5. The index of the root is also called the index of the radical. It is plain that a monomial will be a perfect power, when the numerical co-efficient is a perfect power, and the exponent of each letter exactly divisible by the index of the root. By the definition of a root (Art. 213), we have ("abc . . . ) = abc . . . ; and by the rule for the raising of powers, (√a × √√6 × √c...)" — (√ @)" × (√ õ)" × ('√ c)" ....... =abc......; ... 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 simplest form the imperfect power, 54a13c2. We have 3/54ab3c2 3/27a3b3 x 3/2ac2 = × 2ac2 = 3ab2ac2. 2. In like manner, 3 √/8a2 =2; and √48a5b8c62ab2c √3ac2; 3. Also, 6 192a7bc12/64a6c12 x√3ab2ac2√3ab. 3 In the expressions, 3ab33⁄4√/2ac2, 23⁄4√a2, 2ab2c√3ac2, 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. 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, provided 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 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 to the same index. 3 √2a and (a+b) 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 ab, the value of neither radical will be changed, and the expressions will become 12 12 3√2a = 12√/2+a+ = 12/16a; and √(a + b) = 12 √ (a + b)3. Hence, to reduce radicals to a common index, we have the fol lowing 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, and to raise the quantities under each radical sign to the 6th, 4th, and 3d powers respectively, which gives 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, 3 6 6 6 √2 = √√(2)2 = √√4, and √3 =√√ (3)3 = √√27. 3 5 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√ab and 7 ab; as also, 3a22, and 9c33b2, 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, Again, 3a21√√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√75a = 4b √√3a + 5b √3a = 9b √3a. 3 2. 3√8a3b + 16a1 ·3√/b1+2ab3 = 2a3⁄43√b + 2a − b 3⁄43√b + 2a; 3. 3 3 °3√4a2+2°3√2a = 3 3⁄43√/2a + 2 3⁄43√√2a = 53⁄43√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. Let it be required to multiply a by √. n If we denote the product by P, we have √√ax √√b=P; and by raising both members to the nth power, (√√ a)" × (√b)" = ab = P^ ; and by extracting the nth root, n Vax Vo 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 √. 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, is equal to the nth root of their quotient. |