quadratic, cubic, quadrato-cuhic , and higher Surds, 29 
may be exterminated, any number of surds, whose indices are 
in any manner compounded of the factors 2 and 3, may also 
be exterminated from any equation. 
It may at present suffice to instance, in surds of the 6th, 
and in surds of the 9th power. 
Let ^s/a then, as in cubic surds (3;) 
a, +3^' +3^* 
^/a = 3 ^,/abc ; and +36- vb +36. 
-f 3^* "t" 3^* 
s/c -|- 6 ^/abc = 2y^/abc. Put ^a gb gc = n : then, 
(4O -la. v/'J t”b. v/^’= 21 ^abc ^c-, and, by squaring 
^an + 4a* . a 4- n* — /^bn + 4^" • ^ + H46’ V ^b 
— 44 1 abc + w* — 4^/2 + 46* . c + 84c' — 42^:// ^ab. 
(5;) +b. Tf -4i^ n + \b. n . ^ab: by 
-C. +4<:^ _44iaic 
.-.6a^_2oab-6b^ 
+ . v/^ 6. Put 
,• — ^a^c—d.Z'iabc — %h*c 
restitution, ^ . 
a b — c = r. 
Then r^ i2^6r — 405^6^: 
Square and transpose, r* * 
, — 1231Zfl®’iV. 11,% 1 
r -e,lHaU,. > = 
mensions, free from surds. 
I 
= — 6r — i62cr . ^ab. 
- i2abr* — sys^aber^ Ifsf/baicK 
0, results, an equation of 6 di- 
Let ^\/a -j- ^\/b ^^%/c = 0; then, as in cubic surds (3;) 
^^a -{-^Vr = 2f^\/ abc ; and +6 + ^Wa*b + 3V^6"4- 3 
+ ^ 
^\/ a'c + 6^v/ + S^v^b^'c 4- 3 4- S^^/bc* = 2y^\/abc„ 
