REDUCTION OF SURDS.
236. Transformation of Surds of Any Order into Surds of a Different Order having the Same Value. A surd of any order may be transformed into a surd of a different order having the same value. Such surds are said to be equivalent.
Examples. (1) 3 2= 2^{{1}{3}} = 2^{{4}{12}} = 12 a^4, (2) p a=a^{{1}{p}}= a^{{q}{pq}} =pq a^q.
237. Surds of different orders may therefore be transformed into surds of the same order. This order may be any common multiple of each of the given orders, but it is usually most convenient to choose the least common multiple.
Ex. Express 4 a^3, 3 b^2, 6a^5 as surds of the same lowest order.
The least common multiple of 4, 3, 6 is 12; and expressing the given surds as surds of the twelfth order they become 12 a^9, 12 b^8, 12 a^10.
238. Surds of different orders may be arranged according to magnitude by transforming them into surds of the same order.
Ex. Arrange 3, 3 6, 4 10 according to magnitude.
The least common multiple of 2, 3, 4 is 12; and, expressing the given surds as surds of the twelfth order, we have
3 = 12 3^6 = 12 729, 3 6 = 12 6^4 = 12 1296, 4 10 = 12 10^3 = 12 1000.
Hence arranged in ascending order of magnitude the surds are 3, 4 10, 3 6.
EXAMPLES XXIII. a.
Express as surds of the twelfth order with positive indices :
1. x^{{1}{3}}, 2. a^{-1} \div a^{-{1}{2}}. 3. 4 ax^3 x 8 a^{-1}x^{-2}. 4. {1}{a^{-{3}{4}}} 5. {1}{8 a^{-14}} 6. 6 {1}{a^{-2}}
