30 



Algebra. 



and by direct substitution in the formula we also get 



Fourth case. 



a "\ 



a '' \i 



and by direct substitution in the formula we also get 



= a"'i. 



Thus we see that by using fractional exponents according to 

 the suggestion before obtained, the result of raising any frac- 

 tional power of a to any other fractional power is, in every case, in 

 perfect accord with formula (b). 



30. Examination of formula (c). 



As before, let - and - be any two positive fractions and 



n q n 



P 

 and — any two negative fractions. Then we have four cases to 



consider. 



First case. 



a" —a'J =a" a 'f =^a" '' 



^(y-^'^'^ by Art. 29, second case, and by direct substitution in the formula 

 we also get 



a" -7-«'/=(2" 



Second case. 



a" -^a f =a"a'^ =a" '' , 

 and by direct substitution in the formula we also get 



a" -^a 'J =.a"^ ^ 



