CHAPTER VIII. TRANSFORMATION OF RADICALS. RADICAL EQUATIONS. 123. A RADICAL is an indicated root of a quantity, If the quantity is an imperfect power of the degree indicat ed, it is called a surd. Radicals are divided into degrees, the degree being determined by the index of the root. When the root is indicated by a fractional exponent, the degree of the radical is determined by the denominator of the fractional exponent. *√c, 21√2, z1, (2az3)*, are radicals of the nth degree. 124. Radicals are SIMILAR, when the radical parts are the same; this requires that they be of the same degree, and that the quantities under the radical sign be the same in each. Thus, 73 and 43, are similar PRINCIPLES EMPLOYED IN TRANSFORMING RADICALS 125. Let n denote any whole number whatever, and Raising both members of (1) to the nth power, remem hering that (a)" = a, and ("√√b)" = b, we have, Extracting the nth root of both members of (2), we Things equal to the same thing are equal to each other: hence, equating the first members of (1) and Raising both members of (5) to the nth power, Extracting the nth root of both members of (6), Equating the first members of (5) and (7), It may be shown, as in Art. 116, that when m and n are any positive whole numbers, and a any quantity, we have, From equations (4), (8), and (9), we have the fol lowing principles: 1. The product of the nth roots of two quantities, is equal to the nth root of their product, and the reverse. 2. The quotient of the nth roots of two quantities, is equal to the nth root of their quotient, and the reverse. 3. The mth root of the nth root of any quantity, is equal to the mnth root of that quantity, and the reverse. These principles are of continual application in the transformation of radicals. 126. It has already been shown, that radical quantities can be expressed by means of fractional exponents. The following table of equivalent expressions indicates the different methods of expressing radicals, powers, and reciprocals, according to the conventional principles already alluded to: These are but different methods of writing the same expressions, and we may, at any time, pass from one form of expression to its equivalent. To reduce a radical to its simplest form. 127. A radical is in its SIMPLEST FORM when there is no factor under the sign which is a perfect power. Take the radical, Vays. By factoring, we have, [a3x1у3 = √ a2x+y2 × ay. Hence, from Principle (1), we have, √a3x1y3 = √a2x1y2 × √ay = ax2y Vay. In a similar manner, other radicals may often be simplified. Hence, the rule for reducing a radical to its simplest form. RULE. Resolve the quantity under the radical sign into two factors, one of which is the greatest perfect power of the degree indicated. Extract the required root of this factor, and write it as a factor without the radical sign, leaving the other factor under the sign. Before pronouncing upon the similarity of two radicals they should both be reduced to their simplest form. 3. √(a + x) (a2 — x2) = √(a + x)2 (a − x). |