ALGEBRA, 



nator be multiplied by c, it becomes ?-^ 



bdc 



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- 

 zor, and proceed as in multiplication. 



Let- and -be the two fractions, then 



6 a 

 a c a d a d 



For if = x, and -7 

 b a 



-y, then a = b ^> 



and c 



d y; also, ad b d x, and b c = 



,. a d bdx x a c 

 bdy; therefore = r - r =-= r -f-- 

 b c b dy y b d 

 The rule for multiplying the powers of 

 the same quantity will hold, when one or 

 both of the indices are negative. 



Thus, a m X a n= am n ; for a* X a n 



1 a m 

 = fl X = = a 7 "-* 1 ; in the same 



manner, #3 x x~ 5 = _____ x -2. 

 x> x> 



a~ rni ' n ; because 



Again, a m X 



a X an = - x 



If 7/1= n, am X a, = amm a o. also, 

 a m X a ~ m = = 1 ; therefore a=l ; 



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, a^ -~a~n = am-s -- = a x a 71 



a 

 am+n. 



1 1 a 



7 " -~- a = -- ; -- = = 



a* a a m 

 ^jTn+n. 



Hence it appears,that a quantity may be 

 transferred from the numerator of a frac- 

 tion to the denominator, and the contrary, 

 by changing the sign of its index. Thus, 

 ^m x an _ am am u m Xa~ n 



~bp bp a ' ant a dp Tp 



OJf IirVOIXTIOJST AND EVOLUTION. 



. If a quantity be continu- 

 ally 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-, is called the second 

 power of a,- aXX, or a?, the third 

 power, aXa .(?0> or an * ne w th 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 ar=a*; aX aX 

 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, a m raised to the ?z,th power is a"*". 

 Because a 7 " X m X a 7 *. ...to n factors, by 

 the rule of multiplication, is a mn ; also, 

 ^a~b}n=a bXa bXa 6x&c. to n factors, or 

 a X a + .... to n factors X b X b X A... .to 

 n factors = a^xbn ; and a 1 63 c raised to 

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

 raised to the wth power is a 7 "" ; 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-H be the quantity to be raised 

 to any power. 

 a+b 

 a+b 



or a J -f-2 a b-\-b~ the sq. or 2 d power 

 a+b __ 

 c3-|-2 a 1 b -fa 6 1 



-j- a* 6+2 ab*-\-fr 

 "0+6*3 or o3+3 a- 6+3 a 6H-63 the 3d pr. 



-f a36-j-3a 



a 63 



a 63-ffl* 



a+6H oraH-3a3 6+6a*6-+4 a 63+6* 

 the fourth power. 



If 6 be negative, or the quantity to be 

 involved be a 6, wherever an odd pow- 

 er of 6 enters, the sign of the term must 

 be negative. 



Hence, a 6)* = a* 4 a3 6+6 a* 6 : 

 _ 4 a 63+6 . 



EVOLUTION, or the extraction of roots, 

 is the method of determining a quantity, 

 which, raised to a proposed power, will 

 produce a given quantity. 



