CHAPTER XX RADICALS. IMAGINARY NUMBERS. REVIEW 252. A radical expression is an indicated root of a number or expression. Thus: √2, 7, √10, and √≈ + 1 are radical expressions. 253. Any expression in the form V is a radical expression, or radical. The number indicating the required root is the index of the radical, and the quantity under the radical is the radicand. In 17, the index is 3, and the radicand, 7. 254. When an indicated root of a rational number cannot be exactly obtained, the expression is called a surd. 255. A radical is rational if its root can be exactly obtained, irrational if its root cannot be exactly obtained. Thus: √25 is a rational expression; √10 is an irrational expression. 256. A mixed surd is an indicated product of a rational factor and a surd factor. Thus: 3√5, 4√7x, ab√a + b are mixed surds. 257. In a mixed surd the rational factor is the coefficient of the surd. Thus In 4√5x, 4 is the coefficient of the surd. 258. A surd having no rational factor greater than 1 is an entire surd. Thus: √5 ac is an entire surd. 259. The order of a surd is denoted by the index of the required root. Thus √5 is a surd of the second order, or a quadratic surd. 3/7 is a surd of the third order, or a cubic surd. 260. The principal root. Since (+a)2= + a2 and (−a)2 = + a2, we have √+a2= ± a. That is, any positive perfect square has two roots, one + and the other, but in elementary algebra only the + value, or principal root, is considered in even roots. THE TRANSFORMATION OF RADICALS TO REDUCE A RADICAL TO ITS SIMPLEST FORM 261. A surd is considered to be in its simplest form when the radicand is an integral expression having no factor whose power is the same as the given index. There are three common cases of reduction of surds. (a) When a given radicand is a power whose exponent has a factor in common with the given index. By Art. 244, √a2 = a2 = a+ = √a. α Hence, to reduce a radical to a radical of simplér index: 262. Divide the exponents of the factors of the radicand by the index of the radical, and write the result with the radical sign. Illustration: $8 abx3 = √28a6x3 = 2}a}x} = $/2 a2x. Result. (b) When a given radicand has a factor that is a perfect power whose exponent is of the same degree as the index. By Art. 244, √a2b = (a2b)1 = a3b1 = ab1 = a√b. Hence, to remove from a radicand a factor of the same power as the given index: 263. Separate the radicand into two factors, one factor the product of powers whose highest exponents are multiples of the given index. Extract the required root of the first factor and write the result as the coefficient of the indicated root of the second factor. Illustrations: 1. √12 a3 = √4 a2 × 3 a = 2 avsa. Result. 2. 2√72 a3x+y7 = 2√36 a2x*y* • 2 ay = 2(6 ax2y3) √2 ay (c) When the given radicand is a fraction whose denominator is not a perfect power of the same degree as the radical. If the denominator of a fractional radicand can be made a perfect power of the same degree as the index of the radical, the fractional factor resulting may be removed from the radi cand as in the previous case. By multiplying the denominator by a particular factor we produce the desired perfect power. Multiplying both numerator and denominator by this particular factor introduces 1 under the radical, and the value of the radicand is unchanged. Hence, to reduce any fractional radicand to an integral radicand: 264. Multiply both numerator and denominator of the radicand by the smallest number that will make the denominator a perfect power of the same degree as the radical. By the method of the preceding case remove the fractional power thus formed. The process is the reverse of that of Article 261, Section (a). In general: a√x=a.x1a2x2 = √a2x. Hence, to change a mixed surd to an entire surd: 265. Raise the coefficient of the surd to the same power as the degree of the radical, and multiply the radicand by the result. The indicated root of the product is the required entire surd. Illustrations: 1. 8√5 √32.5= √9.5 = √45. Result. = |