ALGEBRA. 



nator be multiplied by c, it becomes- 



Me 



or ; the quantity which arises from the 

 division of the numerator by c. 



To divide one fraction by another, invert 

 the numerator and denominator of the divi- 

 sor, and proceed as in multiplication. 



Let r-and -7 be the two fractions, then 

 a _ c _ jfl rf _ o_d 

 7~*~d~ b X ~ c ~b~c' 



For if r=x, nd c - = y, then a =c b x, 



and c = dy, also, a d = b d x , and b c= 

 ad b d x x n c 



*d#; therefore T- = r-, - -=- r -7-3. 

 PC o dy y rf 



The rule for multiplying the powers of 

 the same quantity will hold, when one or 

 both of the indices are negative. 



Thus, m x a.* = a"" n -, for a"* X """"" 



1 am 



= a X = a* -, m the same 

 an a 1 * 



X3 1 



manner, *3 X x * = =; = x 2 . 

 *5 x 1 



Again, a X <*~ n = a 1 "*" ; b< cause 



111 _ 

 rt n x a = X = x -=a- m+n 

 a m a n amT 



If m=n, <.w X a" 1 =amm =<>; a l SO) 



cm x a"" 1 = = 1 ; therefore n*= 1 ; 

 utn 



according to the notation adopted. 



The rule for dividing any power of a 

 quantity by any other power of the same 

 quantity holds, whether those powers are 

 positive or negative. 



Thus, am-T-o =a m -: -- =am X a* 

 an 



Again, a n>-d = -- f -- = 



1 



ro 



1 



an 



a 



am 



Hence itappears, that a quantitymay be 

 transferred from the numerator of a frac- 

 tion to the denominator, and the contrary, 

 by changing the sign of its index. Thus, 

 am x a"> > a X n~" 



Tip bp un' aflbp~ 6p 



out nrroLurioir AXD ETOLCTIOW. 



If a quantity be conti- 

 nually multiplied by itself, it is said to be 

 involved or raised; and the power to 

 which it is raised is expressed by the 

 number of times the quantity has been 

 employed in the multiplication. 



Thus, axa, or a 1 , is called the second 

 power of a ; ax<JXa,or 3, the third pow- 

 er, aX"....(n), or an, the n h power. 



If the quantity to be involved be nega- 

 tive, the signs of the even powers will be 

 positive, and the signs of the odd power 

 negative. 



For ax a = a 1 ; ax a X 

 a a?, &c. 



A simple quantity is raised to any pow- 

 er, by multiplying the index of every fac- 

 tor in the quantity by the exponent of the 

 power, and prefixing the proper sign de- 

 termined by the last article. 



Thus, <." raised to the * power is a^m f 

 Because m * " x ....to n factors, by 

 the rule of multiplication, is ; also, 

 iifrn=a bxa bxu x&c. to n factors, or 

 v <i -f- a....to n factors X b X. b X 6... .to 

 n factors =<nx^ n ; and a- fa e raised to 

 the fifth power is a 10 b l< > c*. Also, am 

 raised to the 71* power is <mn where 

 the positive or negative sign is to be pre- 

 fixed, according as n is an even or odd 

 number. 



If the quantity to be involved be a frac- 

 tion, both the numerator and denomina- 

 tor must be raised to the proposed power. 



If the quantity proposed be a compound 

 one, the involution may either be repre- 

 sented by the proper index, or it may ac- 

 tually take place. 



Let a-\-b be the quantity to be raised 

 to any power. 



_ 

 2 or a 2 -f-2a b-\-b* the sq. or 3 d power 



- 



a-f-^3 or a3-j-3 a2 A-J-3 

 a+b 



3 d pr 



04-f 3 as 



the fourth power. 



If b be negative, or the quantity to be 

 involved be a b, wherever an odd pow- 

 er of A enters, the sign of the term must 

 be negative. 



Hence, a ^*=a+ 4 a'- b -f 6 a 1 62 



ErotrTios, or the extraction of roots, 

 is the method of determining a quantity, 

 which, raised to a proposed power, will 

 produce a given quantity. 



