NOTE 1. According to the principles of Art. 136, the signs of the answers given above may all be changed, and still be correct. NOTE 2. No binomial can be a perfect square. For the square of a monomial is a monomial, and the square of the polynomial with the least number of terms, that is, of a binomial, is a trinomial. NOTE 3. A trinomial is a perfect square when two of its terms are perfect squares and the remaining term is equal to twice the product of their square roots. For, (a + b)2 = a2 +2ab+b2 (a - b)2 = a2 — 2ab+b2 Therefore the square root of a2± 2ab+b2 is a ±b. Hence, to obtain the square root of a trinomial which is a perfect square, Omitting the term that is equal to twice the product of the square roots of the other two, connect the square roots of the other two by the sign of the term omitted. 12. Find the square root of x2 + 2x + 1. 15. Find the square root of 16y2 + 40 y2+252. ́t yti z NOTE.By the rule for extracting the square root, any root whose index is any power of 2 can be obtained by successive extractions of the square root. Thus, the fourth root is the square root of the square root; the eighth root is the square root of the square root of the square root; and so on. Since, according to the Binomial Theorem, when the terms of a power are arranged according to the power of some letter beginning with its highest power, the first term contains the first term of the root raised to the given power, therefore, if we take the required root of the first term, we shall have the first term of the root. And since the second term of the power contains the second term of the root multiplied by the next inferior power of the first term of the root with a coefficient equal to the index of the root, therefore if we divide the second term of the power by the first term of the root raised to the next inferior power with a coefficient equal to the index of the root, we shall have the second term of the root. In accordance with these principles, to find any root of a polynomial we have the following RULE. Arrange the terms according to the powers of some letter. Find the required root of the first term, and write it as the first term of the root. Divide the second term of the polynomial by the first term of the root raised to the next inferior power and multiplied by the index of the root. Involve the whole of the root thus found to the given power, and subtract it from the polynomial. If there is any remainder, divide its first term by the divisor first found, and the quotient will be the third term of the root. Proceed in this manner till the power obtained by involving the root is equal to the given polynomial. NOTE 1.- This rule verifies itself. For the root, whenever a new term is added to it, is involved to the given power, and whenever the root thus involved is equal to the given polynomial, it is evident that the required root is found. NOTE 2. As powers and roots are correlative words, we have used the phrase given power, meaning the power whose index is equal to the index of the required root, and the phrase next inferior power meaning that power whose index is one less than the index of the required root. 1. Find the cube root of a-3 a5 +5 a3-3a-1. OPERATION. Constant divisor, 3 a1) ao — 3 a5 + 5 a3 — 3 a — 1 (a2 — a — 1 a3a3 a4-a3 The first term of the root is a2, the cube root of a. a2 raised to the next inferior power, i. e. to the second power, with the coefficient 3, the index of the root, gives 3 a1, which is the constant divisor. · 3 a3, the second term of the polynomial, divided by 3 at, gives a, the second term of the root. (a3 — a)3 — ao —— 3 ao +3 at a; and subtracting this from the polynomial, we have - 3 at as the first term of the remainder. 3 a divided by 3 at gives - 1, the third term of the root. (a2—a—1)3 the given polynomial, and therefore the correct root has been found. = = 2. Find the fourth root of 16 x 32 x3 y2 + 24 x2 y1 - 8 x y + y3. OPERATION. 4 X (2x)3 = 32x3) 16 x — 32 x3 y2+ 24 x2 y* — 8 x y + y3 (2 x − y · — 16 x 3. Find the cube root of a3 + 3 a2 b + 3 a b2 + b3 — 3 a2 c - 6 a b c 3 b2 c + 3 ac2 + 3 b c2 4. Find the fourth root of 16 a1 c1. - 32 a3 c3 x + 24 a2 c2 x2 -8 a cx3 +x04. SECTION XVII. RADICALS. 147. A RADICAL is the indicated root of any quantity, as √x, a3, √2, 34, &c. 148. In distinction from radicals, other quantities are called rational quantities. 149. The factor standing before the radical is the coefficient of the radical. Thus, 2 is the coefficient of No2 in the expression 2/2. 150. SIMILAR RADICALS are those which have the same quantity under the same radical sign. Thus, ✔a, 2 ✔a, and xa are similar radicals; but 2a and 2, or 2x and 2x are dissimilar radicals. 151. A SURD is a quantity whose indicated root cannot be found. Thus, 2 is a surd. The various operations in radicals are presented under the following cases. CASE I. 152. To reduce a radical to its simplest form. NOTE. A radical is in its simplest form when it contains no factor whose indicated root can be found. 1. Reduce 75 a2b to its simplest form. OPERATION. √75 a2b√25 a2 × 3 b = √25 a2 × √3b5a3b We first resolve 75 ab into two factors, one of which, 25 a2, is the greatest perfect square which it contains; then, as the root of the product is equal to the product of the roots (Art. 143, Note 2), we extract the square root of the perfect square 25 a2, and annex to this root the factor remaining under the radical. Hence, RULE. Resolve the quantity under the radical sign into two factors, one of which is the greatest perfect power of the same name as the root. Extract the root of the perfect power, multiply it by the coefficient of the radical, if it has any, and annex to the result the other factor, with the radical sign between them. Reduce the following expressions to their simplest form: 16 x2 y232 x1 y = √16 x2 y21-2 x2 y |