Theory of Indices. 27 



25. Again, by the definition of a quantity with an ex- 

 ponent — I, 



and by formula (b) 



\a'^^a" 



I 

 —z=a 



or, taking the reciprocal of both sides, 



- - I 



a " =^. 



.26 Thus we have suggestions of meanings for both positive 

 and negative fractional exponents, and if we introduce fractional 

 exponents into our formulas with the meanings suggested, these 

 formulas will be found to give consistent results, as we shall see. 



27. Before substituting in our formulas it is necessar}- to stop 



and show that, with the meanings suggested, a quantity with a 



fractional exponent has the same value whether the exponent is 



in its lowest terms or not. 



1 

 Let a^"=ji- 



then a=-x'^"=-(x'^ )" 





In a similar manner it may be shown that 



_ '' _'■"' 

 a "=a f" 



28. Examination of formula (a) 



r . p . 

 I^et — and ~ be any two positive 

 71 q 



two negative fractions. Then there are four cases to consider 



IvCt - and ^ be any two positive fractions and and 



71 q ^ ^ 11 q 



