CHAPTER XVII. RADICAL EXPRESSIONS. 253. A radical expression is an expression affected by the radical sign; as Va, V9, Va2, Va + b, √32. 254. An indicated root that cannot be exactly obtained is called a surd. An indicated root that can be exactly obtained is said to have the form of a surd. The required root shows the order of a surd; and surds are named quadratic, cubic, biquadratic, according as the second, third, or fourth roots are required. The product of a rational factor and a surd factor is called a mixed surd; as 3√2, b√a. The rational factor of a mixed surd is called the coefficient of the surd. When there is no rational factor outside of the radical sign, that is, when the coefficient is 1, the surd is said to be entire; as √2, Va. 255. A surd is in its simplest form when the expression under the radical sign is integral and as small as possible. Surds are said to be similar if they have the same surd factor when reduced to the simplest form. NOTE. In operations with surds, arithmetical numbers contained in the surds should be expressed in their prime factors. Reduction of Radicals. 256. To reduce a radical is to change its form without changing its value. CASE 1. 257. When the radical is a perfect power and has for an exponent a factor of the index of the root. We have, therefore, the following rule: Divide the exponent of the power by the index of the root. 258. When the radical is the product of two factors, one of which is a perfect power of the same degree as the radical. Since Warb = Van x bab (§ 224), we have 1. √ab = √a2 × √b = a√b; X 2. √108 = √27 × 4 = √27 × √4 = 3√4; 3. 4√72 a2b8 = 4√36 a2b2 × 2b = 4√36 a22 × √2b = 4 × 6 ab √2b = 24 ab √2b; 4. 2√54a+b=2√27 a3 × 2 ab = 2√27 a3× √2 ab We have, therefore, the following rule: Resolve the radical into two factors, one of which is the greatest perfect power of the same degree as the radical. Remove this factor from under the radical sign, extract the required root, and multiply the coefficient of the surd by the root obtained. CASE 3. 259. When the radical expression is a fraction, the denominator of which is not a perfect power of the same degree as the We have, therefore, the following rule: Multiply both terms of the fraction by a number that will make the denominator a perfect power of the same degree as the radical; and then proceed as in Case 2. CASE 4. 260. To reduce a mixed surd to an entire surd. Since a b = Va" × Vb = Vab, we have 1. 3√5 = √32 × 5 = √9 × 5 = √45; 2. a2b√bc = √(a2b)2 × be = √a1b2 × bc = 3. 2x √xу= √(2x)3 × xy = 4. 3y2√x3 = √(3y2)* × x3 = 3/8 x8 xxy = $8x+y; We have, therefore, the following rule: Raise the coefficient to a power of the same degree as the radical, multiply this power by the given surd factor, and indicate the required root of the product. V5=3437= V32 = √9. We have, therefore, the following rule: |