36 Algebra. 



3. Definitions. In the following pages, by the word Radical 

 may be understood the indicated root of an expression, whether 

 that root is indicated by the ordinary radical sign or by a frac- 

 tional exponent. 



By the Index of a radical may be understood either the number 

 written in the angle of the radical sign or the denominator of the 

 fractional exponent. 



A multiplier written before a radical will sometimes be called 

 the co-efficie7it of the radical. 



A Simple radical is the indicated root of a rational expression. 



A Complex radical is the indicated root of an irrational 

 expression. 



A 7nonomial Surd is the name applied to the indicated root of 

 a commensurable number, when that root cannot be exactly taken ; 

 as n/|, or V3. 



If all the irrational terms in a binomial or polynomial are surds, 

 it is called a binomial ox polynomial surd, as the case may be. 



It should be noticed here that we make a distinction between the terms 

 irrational expression and surd, a distinction which is not commonly made, 

 the two terms being generally defined as identical. According to the above 

 definition, ^4 -^l-^-^^ v^^lj, V" are not surds. But they are irra- 

 tional by the definition of I, Art. 3. This limited meaning of the word surd is 

 convenient and is growing in use. It is found in both Aldis' and Chrystal's 

 algebras. 



Radicals are said to be Similar when they have the same index 

 and the expressions under the radical signs are the same ; that is, 

 two radicals are similar when they differ only in their coefficients. 

 Such are ^^ ab and m^/ ab\ also f^y and f^y. 



4 , Definition. For a radical to be in its simplest form it is 

 necessary (i) that no factor of the expression under the radical 

 sign is a perfect power of the required root ; (2) that the expres- 

 sion under the radical sign is integral ; (3) that the index of the 

 radical is the smallest possible. 



It will be seen from the following pages that every simple radi- 

 cal can be placed in this form without changing its value. The 

 transpositions necessary to effect the reductions depend upon cer- 

 tain principles, or theorems, established in the last chapter, which 

 we collect here for reference. 



