i8 



Algi:bra. 



Now, in general, if n and r are positive whole numbers and 



a" _aaaa . . . to n factors, 

 a'' aaa . . . . to r factors, 

 and the r a's in the denominator will cancel r of the ^'s in the 

 numerator and leave (?i — r) a's iri the niDnerator. 



.'. a"-r-a''=a"~''. (c) 



This formula will be referred to as formula (c). 



Again, n and r still being positive whole numbers, if 7i<ir, we 



have, as before, 



a'' _aaaa . . . to n factors, 



a'' aaa . . . . to r factors, 

 but here the 7i a's in the numerator will cancel ?i of the «'s in the 

 denominator and leave (r—?z) a's m the denominator. 



a"-^a''=-—-. when n<^r. (d) 



This formula will be referred to as formula (d). 



8. We may also find the quotient of the same powers of dif- 



ferent quantities. 



and 



and so in general. 



aaa 

 '' b b b 



a" a a a a ^ a^ 



b^' 



bb 



aaa 



'"bVb 



b b \b 

 aaa 

 '"b b b^ 



aaa . . . to n factors. 



b" bbb . . . to 7^ factors, 

 . . to 71 factors each of which equals -, 

 a 



9. In the case just considered we have the sa7ne power, but the 

 quantities a and b may be different, but in Art. 7 we have the 

 same qiia7itities , but the powers may have been different. The 

 student should not confuse these two cases. 



