By         — 3a3         263          a2  (a—h-\-y)*

QUEST. — What  is  the  product  of  the  sum  and  difference  of  two  quan-
tities equal  to  ?

10


110  ALGEBRA.  [Sect.  VIII.
5.  The  quotient  of  a5  divided  by  a3,  is  ^.      But  this   is

equal  to  a2.     For,  in  the  series

a+S  a+3,  a+2,  o+1,  a°,  a'1,  a"2,  «'3,  a'S  &c.
if  any  term  be  divided  by  another,  the  index  of  the  quotient
will  be  equal  to  the  difference  between  the  index  of  the  divi-
dend  and  that  of  the  divisor.
Thus  8«4«'=  -«2-     And

aaa  a

193.  Hence  we  deduce  the  following

GENERAL     RULE     FOR     DIVIDING     POWERS.
I.  A  power  may  be  divided  by  another  power  of  the  same
root,  by  subtracting  the  index  of  the  divisor  from  that  of  the
dividend.
II.  If  the  divisor  and  dividend  have  co-efficients,  the  co-
efficient of  the  dividend  must  be  divided   by  that  of  the  divi"
sor.     (Art.  96.)

III.  If  the  divisor  and  dividend  are  both  compound  quan-
tities^ the  terms  must  be  arranged,  and  the  operation  conduc-


ted in  the  same  manner,  as  in  simple  division  of  compound
quantities.     (Art.  107.)

6.  Thus  y^-^-yz—y^-z—y^.     That  is,  ^  —y.
J \J

7.  Divide  an+l  by  a.  8.  Divide  zn  by  xn.
9.  10.  11.  12.  13

Divide      y2m  b6  8an+m          «n+3

By  ym  b3  4am  a2

194.  The  rule  is  equally  applicable  to  reciprocal  powers.
Q.UEST. — How  is  a  power  divided  by  another  power  of  the  same
root?  If  they  have  co-efficients,  how  proceed?  When  the  divisor
and  dividend  are  both  compound  quantities,  how  ?  Is  this  rule  appli-
cable to  reciprocal  powers  ?

°'


Arts.  193-194.a.  ]  POWERS.  Ill

Thus  the  quotient  of  «"5  by  a"3,  is  a"2.
