SECOND TRANSFORMATION. - To introduce a factor under the radical sign. 127. Take the example, 4a2c: Since 4a = = √16a2, the given expression may be written, 4a√//2c = √16a2 × √2Zc, which, by principle (1), may be written, 4a√2c = √16a2 × 2c = √/32a3c. In like manner, any factor without the radical sign may be introduced, as a factor, under the radical sign; hence, the following RULE. Raise the factor to the power indicated, and write it as a factor under the radical sign. EXAMPLES. Transform the following radicals by introducing the coefficients, as factors, under the radical signs: This transformation is used in finding the numerical values of radicals. Thus, it is easier to find the cube root of 432 than to find 12 times the cube root of, to which it is equivalent. THIRD TRANSFORMATION.-To change the index of a radical. 128. Take the radical, 3a: Since 3a is equal to √3a × 3a, or √9a2, we have, Here, a radical having an index 3 has been transformed to an equivalent radical, having an index 6. Since we may proceed in like manner in all similar cases, we have the following Principles. 10. The index of any radical may be multiplied by any number, provided we raise the quantity under the radical sign to a power whose exponent is the same number. 2°. The index of a radical may be divided by any number, provided we extract that root of the quantity under the radical sign whose index is the same number. If a radical is expressed by means of a fractional exponent, we may proceed as follows: (a) is evidently equal to (a)3, since ‡ = 3. Also, (a) is equal to (a), since is equal to †; These are the same results as obtained by the rule. From what precedes, we have the following Principle. 3°. Both terms of a fractional exponent may be multiplied, or divided, by the same quantity without changing the value of the radical. EXAMPLES. Verify the following equations: 1. √3 + √4 + √5 = ✩/9 + √16 + 1/25. 2. √25 - √27-49 3. 3a2-2ax+x2 - 2√/8a3x3 = 3√a―x — 2√/2ax. 4. Va-x-√a+x= √ a2−2ax+x2 — √ a2+2ax+x2. 5. (ax)*+ (by)‡ = (ax)2 + (by)3. 6. (z)‡ — (y)* = (x)* — (y)*. FOURTH TRANSFORMATION. common index. To reduce radicals to a 129. Radicals may be reduced to a common index by means of the preceding principles. Let it be required to reduce the radicals, a, b, and Vc, to equivalent ones having a common index: here, the least common multiple of the indices is 12; reducing each to the index 12, by the foregoing principles, we have, = 12/64, and Ve= 12/08. Since all other cases may be treated in the same way, we have the following RULE. Find the least common multiple of the indices, and reduce each radical to that index. If the radicals are expressed by fractional exponents, we have simply to reduce these exponents to a common denominator. EXAMPLES. 1. Reduce 2, (3), (a), and (b), to a common inAns. (212), (34)1, (a), and (68)15. dex. |