}6^ ARITHMETIC. 



2. Find the first figure of the root by trial; and subtract 

 Its power from the given number. 



3. To the remainder bring down the first figure in the 

 next period, and call it the di-vide7id, 



4, Involve 



Extract the cabe root of a^ -\-6a^ — 40^^4-9^^ — ^4* 

 a^ -^Ga^ — 40^?^ -\-Q6a — 64(^*4- 2a — 4 



^a^)6a^ [J^2a 



a«4.6«^4-i2a^ + 8a5=a^ + 2a 



^^ 4-2^ X3=3«*'^ + 1 2^^ + 12^*) — I2dr^ — 48/2^+96^ — 64 ( — /^ 

 a^ -\' 6a^ — 40^' + 9^^ — 64 = ^ " 4- 2a — 4 



When the index of the power, whose root is to be extracted^ 

 is a coiTipositc number, the following rule will be serviceable : 



Take any tv;o or more indices, whose product is the given in- 

 dex, and exiract out of the given number a root answering to one 

 of these indices ; and then out of this root extract a root answer- 

 inor to another of the indices, and so on to the last. 



Thus, the fourth root = square root of the square root. 

 The sixth root == square root of the cube root, Sec. 

 The proof of all roots is by involution. 



The following theorems may sometimes be found useful in ex- 

 tracting the root of a valvar fraction : \/ — zzY zz ^_ — =: 1- ; 



"^ ' " b ^b b ^^ 



or, universally, ^1 ^Jl ah 



i 



^l in 



7F\ 



