223. The two methods of indicating roots may be illustrated by equivalent expressions, as follows: Vam, or a, denotes the square root of am; 224. A Surd is a root which cannot be exactly obtained; as √2, Va2, or Va2-2ab. A surd is called an irrational quantity, while a root which can be exactly obtained is called a rational quantity. A root will be rational when the given quantity is a perfect power corresponding in degree to the required root; otherwise it will be a surd. The root of a number which is an imperfect power, may always be obtained approximately. Thus, √6 is a surd; but we have √6 = 2.44, nearly; for (2.44)2 = 5.9536. 225. An Imaginary root is one which is known to be impossible on account of the sign of the given quantity. Thus, the square root of a2, or √ a2, is impossible, since no quantity raised to the second power will produce — a2. A root which is not imaginary is said to be real. ROOTS OF MONOMIALS. 226. It has already been shown that the root of a simple algebraic quantity may be expressed by dividing the exponent of the quantity by the index of the required root (222). And it is evident that if the exponent of the quantity will not exactly contain the index of the required root, the result must be a surd. 227. We have seen that a quantity composed of several factors, may be raised to any power by involving each factor separately to the required power (208). Conversely, we may obtain the root of a quantity by extracting the root of each factor separately. Thus, if abc.... k represent the product of any number of factors, then abc....kva V b V c.... Vk; The nth root of the product of two or more factors is equal to the product of the nth roots of the factors. 228. There are certain properties of roots which depend upon the law of signs in involution: 1. Every odd root of a quantity is real, and has the same sign as the quantity itself. For, any positive quantity raised to an odd power is positive; and any negative quantity raised to an odd power is negative (210). 2. Every even root of a positive quantity is real, and may be either positive or negative. For, either a positive or a negative quantity raised to an even power is positive (210). 3. Every even root of a negative quantity is imaginary. For, no quantity, whether positive or negative, raised to an even power, will give a negative result. 229. From the principles now established, we have the following rule for extracting the roots of monomials: RULE.-I. Extract the required root of the numeral coefficients for a new coefficient. II. Divide the exponent of each literal factor by the index of the required root. III. Prefix the double sign, ±, to all even roots, and the minus sign to the odd roots of a negative quantity. NOTES.-1. When the required root of any factor is a surd, it may be indicated either by a fractional exponent, or by the radical sign. 2. The root of a fraction may be obtained by taking the root of the nu merator and denominator separately. 11. Find the 5th root of —32x1y1. Ans. —2x2y*, or —2x2√/y+. 25. Find the square root of x2y1 (x—y)2. Ans. ±(x2y2—xy3). SQUARE ROOT OF POLYNOMIALS. 230. To deduce a rule for the extraction of the square root of a polynomial, let us first observe how the square of any binomial, as a + b, is formed. We have (a + b)2 = a2 + Qab + b2. And the last two terms may be written as follows: (2a + b) b. may be OPERATION. a2+2ab+b2 a+b a2 Let us now consider how the process of involution reversed, and the root, a + b, derived from the square. Extracting the square root of a2, we obtain a, the first term of the root. Taking a2 from the whole expression, we have 2ab+b2, or (2a + b) b, for a remainder. Dividing the first term of this remainder by 2a, as a partial divisor, we obtain b, which we place in the root, and also at the right of the 2a to complete the divisor, 2ab. Multiplying the complete divisor by b, and subtracting the product from the dividend, we have no remainder, and the work is finished. 2a+b 2ab+b2 By the same process continued, we may extract the square root of any quantity that is a perfect square. To establish the rule in a general manner, let a+b+c+d.... represent any polynomial. By a previous article, the square o this polynomial consists of the square of each term, together with twice the product of each term by the sum of all the terms which follow it (217); and the square may be written as follows: a2+2ab+2ac+2ad.... +b2+2bc+2bd .... +c2+2cd....+ď2.... And it is evident that if the root, a+b+c+d...., is arranged according to the powers of some letter, the square will also be arranged according to powers of the same letter. We may now derive the root from the square, in the following manner: OPERATION. \a+b+c+d...., root. a2+2ab+2ac+2ad....+b2 +2bc+2bd....+ c2 + 2cd . . . . + ď2 . . . . .... We find a as in the former example, and take its square from the whole expression. We then divide the first term of the remainder by 2a, and write the quotient, b, in the root, and also in the divisor. We then multiply the complete divisor by b, subtract the product from the first remainder, and thus obtain a new dividend. Then writing 2a + 26 for a partial divisor, we find c in the same manner as we found b; and thus we continue till the work is finished. If we examine the several subtrahends, taking the terms diagonally in the operation, we shall find a2, 2ab, 2ac, 2ad, etc.; b2, 2bc, 2bd, etc.; c2, 2cd, etc.; d2, etc. That is, we have, in the operation, the square of each term of the root, together with twice the product of each term by all the terms which follow it. Thus we have exactly reversed the process of forming a polynomial square. Hence the following general RULE.-I. Arrange the terms according to the powers of some letter, and write the square root of the first term for the first term of the root. II. Subtract the square of the root thus found from the given quantity, and bring down two or more terms for a dividend. III. Divide the first term of the dividend by twice the root already found, and write the result both in the root and in the divisor. IV. Multiply the divisor, thus completed, by the term of the root last found, subtract the product from the dividend, and proceed with the remainder, if any, as before. NOTE. According to the law of signs in evolution, every square root obtained will still be a root, if the signs of all its terms be changed. |