32 Algebra. 



33. By means of the meanings now given to negative and 



fractional exponents it is easy to see that the formula 



a"b"c" . . . =(a d c . . , )" 



holds whether ;/ is positive or negative, integral or fractional. 



I — ^ 



First, let ?^= ^, a positive fraction, and let a''=x and d''=y; 



.'. a=x" and b=y" 

 1 1 

 then a'b''=xj\ 



and ab=xy'=(xy)'\ 



.'. (ab)'' =xj', 



_1 1 _L 



,\ a '■ b'' = fab)'' . 



Multiply both sides by c ' and we get 



1 1 1 1 I A 



a'' b'c'' = (ab) '' C = (abc) '' , 



and so on, evidently, for any number of factors. 



This is quite an important formula, stated in words it is, 

 The product of the r th roots of several quantities equals the rth 

 foot of their product. 



Second, let u=—r, a negative quantity, either integral or frac- 

 tional, then 



a"^ b'^ (ab) 

 Similarly a'"'b~''c~''= (abc)~'', 



and so on, evidently, for any number of factors. 



34. The formula 7„= 't[ also holds good whether ii is posi- 

 tive or negative, integral or fractional. 



1 1 J 



First, let ^2=^, a positive fraction, then -—a''\~,\ — -j - - 



b'' 

 Stated in words, this is. 



The quotient of the rth roots of two quantities equals the rth root 

 of their quotieyit. 



