231. The COEFFICIENT of a radical is the quantity or factor prefixed to it. Thus, in 22bc3, and a (c + d), 2 and a are the coefficients. 232. The DEGREE of a radical is denoted by the index of the radical sign, or by the denominator of the fractional exponent. Thus, √a, √y, (abc), are radicals of the second degree; ✅x2, b3, (2 u2x23y)3, are radicals of the third degree; 1 ac, 3m, (a+b)”, are radicals of the nth degree. 233. SIMILAR RADICALS are those of the same degree, with the same quantity under the radical sign. Thus, 5ax and 7ax are similar radicals; and also a y 1 and cy". 1 REDUCTION OF RADICALS. 234. REDUCTION OF RADICALS is the process of changing their forms without altering their values. 235. The reduction of radicals depends upon the general principle, that The root of any quantity is equal to the product of the like roots of its several factors (Art. 212). CASE I. 236. To reduce radicals to their simplest form. A radical is in its simplest form, when it has under the sign no factor which is a perfect power of the same degree as the radical. 1. Reduce 54 a5 x to its simplest form. OPERATION. 54 ax = 27 a3 x3 × 2 a2 x 27 a3 3 2 a2 x = 3 ax2 a2 x We resolve the quantity under the radical sign into two factors, one of which, 27 a3 x", is the greatest perfect cube contained in the radical quantity, and the other the surd 2 ax. But 27 a 3 ax; consequently, by 3 = the general principle (Art. 235), the given radical is equal to the product of 3 ax and 2 ax, or 3 ax✅ 2 a2 x. Hence the RULE. Resolve the quantity under the radical sign into two factors, one of which shall contain all the perfect powers of the same degree as the radical. Extract the required root of this factor, multiply the root by the coefficient of the radical, and prefix the product to the other factor, with the radical sign between them. When the given radical has no coefficient expressed, the coefficient 1 is always understood. EXAMPLES. 2. Reduce 27 a to its simplest form. Ans. 3 ax23 ax. 3. Reduce (189 at b3 c2) to its simplest form. Ans. 3 ab (7 ac2). 4. Reduce 96 a2 x3 to its simplest form. α Ans. 4 ax 6 x. 5. Reduce a+b to its simplest form. 6. Reduce 2c768 to its simplest form. Ans. 8c 3. 7. Reduce a3 — a2b to its simplest form. Ans. aa- b. 8. Reduce (ax2- 6ax+9a) to its simplest form. Ans. (x-3) a. a3 9. Reduce√ (x2 — y2) (x+y) to its simplest form. Ans. (x+y)xy. 237. When the given radical is in a fractional form, we may transform it in such a manner as to make the denominator a perfect power of the degree indicated. Then, the surd factor will be an entire quantity. 238. To reduce a rational quantity to the form of a radical. 1. Reduce 2 x2 to the form of the cube root. OPERATION. 2x2 = (2 3) = (8 = 8. Here, 2 x2 cubed is 8x; and, reverse of involution, 826 written root indicated, gives (8x), or form. Hence the since evolution is the under the sign of the 8x6, as the required RULE. Raise the quantity to the power indicated by the given root, and write it under the corresponding radical sign. EXAMPLES. 2. Reduce a +x to a radical of the nth degree. 3. Reduce y2 to the form of the square root. Ans. ✔✅ (x — y2)2. 239. A coefficient, or a factor of a coefficient, of a radical may be placed under the radical sign, by raising it to the power indicated by the radical, and multiplying the quantity already under the sign by the result. 6. Introduce the coefficient of 2 x y 5 under the radical sign. 2xy√5 = √ (2 x y)2 × 5 7. Reduce x (2 a — x) to a radical without a coefficient. Ans. (2 a x2 — x3)*. 8. Introduce the factor 3 of the coefficient of 3 a m under the radical sign. Ans. a 3" m. |