CHAPTER II. 



THEORY OF INDICES. 



1. By the definition of a power of a number, a" equals the con- 

 tinued product of ;/ factors each equal to a. 



a"=a a a a to n factors, 



71 being a positive whole number. 



2 . To find the product of two powers of the same letter. 



a^=.a a «, 



.-. a?a'^=a a a a a=a^. 

 Again, a>=a a a a a, 



a^=^a a a, 

 .'. a^a^=a a a a a a a a=^a^. 

 In each case it is to be noticed that the exponent of the pro- 

 duct equals the sum of the exponents of the two factors. 

 In general, if n and r are a?iy positive whole Jiumbers, 



a"^=a a a a to w factors, 



a''=^a a a . .. . . . . to r factors. 



.-. a"a''—a a a a to (n^r) factors =«""'"^ 



In the present chapter the formula, 



a"a''=^a"^'', (a) 



will be referred to as formula a. 



This may be expressed in words thus — 



The product of tivo powers of a qiiantitiy is equal to that quantity 

 with an expo?ient equal to the sum of the exponents of the twofac- 



t07'S. 



3. We may also find the product of the same powers oi dix^^x^wt 

 quantities. 



a'b'^a a b b=(ai)(ab) = (aby; 



also a^b^=a a a b b b=(ab)(ab)(ab)=(ab)^. 



And so in general, 



a"b"=a a a . . . to n factors X/^ b b . . . to w factors, 

 = (ab)(ab)(ab) . . . to ;2 factors, each of which is ab, 



= (ab)\ 

 .-. a"b"==(ab)". 



