CHAPTER VIII. RADICALS. I. TRANSFORMATION OF RADICALS. Definitions. 121. A radical is an indicated root of a quantity. Thus, V3, V16, V16, 3ņa, are radicals. A surd is an indicated root of an imperfect power of the degree indicated. Thus, V3 is a surd. The coefficient of a radical is the factor without the radical sign. Thus, in the expression, 313, 3 is the coefficient; in the expression V3, 1 is the coefficient. 122. Radicals are of different degrees, the degree being determined by the index of the radical. Thus, V3 is a radical of the second degree; V7 is a radical of the third degree, and Va is a radical of the oth degree. 123. Radicals are similar when the radical parts are alike, that is, when they are of the same degree and when the quantities under the radical sign are the Thus, 30 ab and 5 ab are similar. same. Notation. 124. It has already been explained that radical quantities can be written by means of fractional exponents. The following table indicates the conventional methods of expressing radicals, powers, and reciprocals : The numerator of a fractional exponent indicates the power to which the quantity is to be raised, the denominator shows what root of that power is to be taken, and the sign of the exponent tells us whether the result is to be regarded as a factor, or as a divisor. Thus, the expression, c#, shows us that æ is to be cubed, that the fourth root of this cube is to be extracted, and finally that reciprocal of the result is to be taken. Demonstration of Principles. 125. Let n denote any whole number whatever, and Raising both members of (1) to the ntr power, remembering that (Ma)" = a, and (VT)" b, we have, Extracting the nth root of both members of (2), we have, Vab (3) = p . Things equal to the same thing are equal to each other; hence, equating the first members of (1) and (3), we have, Va x o = Vab (4) Again, assume ♡a (5) ♡ = a n 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), From the principle demonstrated in Art. 112, we have, (9) m V Va = "va From equations (4), (8), and (9), we have the following principles : 1°. The product of the nth roots of two quantities, is equal to the nth root of their product, and the reverse. 2o. The quotient of the ner 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 used in the transformation of radicals, that is, in changing their forms, without affecting their values. To reduce a radical to its FIRST TRANSFORMATION. simplest form. 126. A radical is in its simplest form when there is no factor under the sign which is a perfect power of the degree indicated. Take the radical, Va8z4y8: Factoring the quantity under the radical signs, we have, Vasx4y8 va x4y2 x ay. In a similar manner, other radicals may be simplified; hence, the following 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 the result as a factor without the radical sign, leaving the other factor under the sign. Before pronouncing on the similarity of two radicals they should both be reduced to their simplest form. 3. V(a + x) (a” – x2) = v (a + x)" (a — «). Ans. (a + x) Va — X. 4. 2a4r + acid = VaR(2ax + 22). Ans. a 2ax + x?. It will often be advantageous to multiply both terms of a fraction by such a quantity as will make the denominator a perfect power of the degree indicated, in which case, the factor remaining under the sign will be entire. Thus, in the 5th example, |