56 AN ESSAY ON THE ROOTS OF INTEGERS, 



(10.) It would take up a great deal of room to go on demonstrating 

 the following propositions generally for every value of n. It will be much 

 shorter, and equally legitimate, to fix upon an individual index, and 

 demonstrate the extraction of that Root, and then the demonstration 

 may be easily extended to any other Power whatever, by means of the 

 Binomial Theorem. In doing this, I must endeavour not to assign the 

 value of the index n so high as to render the process unnecessarily prolix 

 and cumbersome, and, on the other hand, it must not be taken so low as to 

 render its extension to higher values, obscure and unsatisfactory. Be- 

 tween these two extremes, I shall chuse the number 6, and, making nz=6, 

 shall proceed to demonstrate the extraction of the 6th Root. 



(11.) By the Binomial Theorem (x+z) 6 = x 6 + 6x 5 z + 15x*z z + 20x*z* 



+ 15x z z* +Gxz s +z 6 . 



i; — 



and hence x+z =\x 6 -+- 6x 5 z -f 15x 4 z" + Wx 3 z 3 -f 15x*z 4 + 6xz 5 +z 6 . 



Now let s and t be any real numbers, and there be given the number 



s 6 +t, in which s is knoAvn, then if there can be found a number such, that 



6* s 5 X that found number 



15- 



s 4 



s z 

 s 



X 

 X 

 X 

 X 



that found number | 



20- 



that found number | 



15- 



that found number | 



6: 



that found number | 



< 





that found number | 



When all added together, the 

 sum should be == t, then is 

 s + that found number, the 

 6th Root of s 6 + t. 



For let this found number be u, then evidently the above expression 

 becomes 



6s 3 x « + 15s 4 X u- + 20s 3 x w 3 + 15s- X W -f Gs x « 5 + « 6 - = *• 

 and then s 6 + 6s'u + 15sV + 20*V + 15*V + 6su" + u c ' = s c + t 



x I A 



and then | s 5 + t - * s r ' + 6s'u + 15sV + 20sV + 15sV + Qsw' + « 6 = s + u 



as above by Binomial Theorem. 



