ALGEBRA. 



Since the n^ power of a m is a", the 

 31 th root of a m n must be a m ; i. e. to ex- 

 tract any root of a single quantity, we 

 must divide the index of that quantity by 

 the index of the root required. 



When the index of the quantity is not 

 exactly divisible by the number which 

 expresses the root to be extracted, that 

 root must be represented according to 

 the notation already pointed out. 



Thus the square, cube, fourth, n* root 

 offl^-f x z , are respectively represented by 



(a> + *\H, (a 1 4- **)* (* + * l )fc 

 (a 1 -f" xl )n; the same roots of-; - ;, or 

 (aH-x 1 ) 1 ,arerepresentedby(a i -f.r 1 ) i 

 ** * s * I ~ 



If the root to be extracted be express- 

 ed by an odd number, the sign of the root 

 will be the same with the sign of the pro- 

 posed quantity. 



If the root to be extracted be express- 

 ed by an even number, and the quantity 

 proposed be positive, the root may be 

 either positive or negative. Because 

 either a positive or negative quantity, 

 raised to such a power, is positive. 



If the root proposed to be extracted be 

 expressed by an even number, and the 

 sign of the proposed quantity be negative, 

 the root cannot be extracted ; because no 

 quantity, raised to an even power, can 

 produce a negative result. Such roots are 

 called impossible. 



Any root of a product may be found by 

 taking that root of each factor, and mul- 

 tiplying the roots, so taken, together. 



Thus, (a 6)"ri=aX 6"~; because each 

 of these quantities, raised to the nd pow- 

 er, is a b. 



C a_f2 a b-}- b* (a-f b 



a* 



a b+b* 

 2 a b+b* 



Since the square root of a*-f 2 a b-\-b z 

 is a-f , whatever be the values of a and 

 b, we may obtain a general rule for the 

 extraction of the square root, by observ- 

 ing in what manner a and b may be deriv- 

 ed from a 1 +2 a b+b\ 



Having arranged the terms according 

 to the dimensions of one letter, a, the 

 square root of the first term a 1 is a, the 

 first factor in the root; subtract its square 

 from the whole quantity, and bring down 

 the remainder 2 a 6-f b 1 ; divide 2 a 6 by 

 2 , and the result is b, the other factor in 

 the root ; then multiply the sum of twice 

 the first factor and the second (2<i-f 6), 

 by the second (6), and subtract this pro- 

 duct J(2 a -f 6 1 ) from the remainder. 

 If there be no more terms, consider a-\-b 

 as a new value of a ; and the square, that 

 is a z -f 2 a b-\-fr, having, by the first part 

 of the process, been subtracted from the 

 proposed quantity, divide the remainder 

 by the double of this new value of a, for 

 a new factor in the root ; and for a new 

 subtrahend, multiply this factor by twice 

 the sum of the former factors increased 

 by this factor. The process must be re- 

 peated till the root, or the necessary ap- 

 proximation to the root, is obtained. 



Ex. 1. To extract the square root of 

 2-f2 ab+b*+ 2 a c-f 2 b c+c a . 

 a>-f 2 a 6+* l -f 2 a c-f 2 b c-f c 1 (a 



2 



a b+b* 

 2 a b+b* 



In 



i JL - 

 , then a" X a=a ; and in the 



2 a c-f 2 b c+c- 

 2 a c-f 2 b c-f c> 



a 2 a x -f 



same manner ax <?= a . Ex ' \ To e ^aetthe square root of a>. 



Any root of a fraction may be found by ^"^"^ 

 tpkingthatrootboth of the numerator and 



denominator. Thus, the cube root of ^ is 



2 /,x i 



^I,ora|x^ |; and f J 



*| \*> 



= ?; 



3 / * 



4( a -S 



, ?i 



or a 



To extract the square root of a compound Ex. 3. To extract the square root of ] 

 quantity, 4-0-. 



