146 Algebra. 



,• A power of a given base may have more than one value, as for 



instance, (^25/'* = ±5, but any commensurable pozver of a base has 

 aniORir its values one which is positive. 



For any integral power of a positive base is evidently positive, 

 and au}^ root of a positive base has among its values one which is 

 positive, and since ^ny pozver of this root is positive so any posi- 

 tive or n^gaJdvc fractio7ial power of a positive base has among its 

 values one which is positive, and as a negative power of a base is 

 the reciprocal of a positive power of the same base, any negative 

 fractional power of a positive base has among its values one which 

 is po'sitive. This positive value is all that is considered in the 

 present discussion. So that whenever we deal with a quantity 

 like a in the present chapter, both a and a"- are positive. ThCvSe 

 restrictions must not be lost sight of. 



16. Theorem. If x and y are any tzl'o commensurable 

 numbers zvhere y is greater than x, then a^ is greater than a"^^ if a is 

 greater than unity, and a^ is less than a"" if a is less than loiity. 



First Case. When a is greater than unity. 



and since jF>-f, y—x is positive, and therefore ^'~' is greater than 

 unity, for a positive power or root of a quantity greater than unity 

 is itself greater than unity. 



Hence, ar' -7- a' > i 



hence a^^a' . 



Second Case. Where a is less than unity. 



As before a^' -^a"" =^-'~' 



and «'"''< I, for a positive power or root of a quantity less than 

 unity is itself less than unity. 



Therefore a^' -j-<2^ <^ i 



and hence a^' <«^ . 



Therefore if a is greater than unity, the greater x is the greater 

 is a" , or in other words, if a is greater than unity, a"" increases as 

 x increases, and \{ a is less than unity, a"" decreases as a" increases. 



17. Consider a quantit}^ q intermediate in value between x 

 and J', then if a is greater than unity a ' <a'^ <<2' , and if a is less 



