142

CUBE BOOT OF A BINOMIAL.

TRACT 32.

root of the former, will be the other member of the rootsought.” But this rule will perhaps be better understoodin characters thus : Let in be one member of the given bi-nomial, whose cube root is sought; and let it be divided intothe two parts a 3 and 3b, so that a 3 3b be = in ; then is

the cube root of the proposed quantity, if it has a

root. Thus, in the quantity ^/108-f- 10, taking the termVl08 for in, this divides into the two equal parts V2l and.y/27, making <z 3 =. x /27, and 3b = a /27 also ; hence a rr.y/3,

and b=./3 also; consequently a + V b - =\/3 +vAf or

V3 + 1, for the cube root of the binomial sought. Again,taking the term 10; this divides into 1 and. 9, where

a 3 = 1 or a = 1, and 3b = 9, or b — 3 ; therefore a + y/-^

becomes 1 + v/3 for the cube root of y/108 + 10, the sameas -before.

“ And thus,” Tartalea adds, “ we may know whetherany proposed binomial or residual be a cube or a noncube;for if it be a cube, the same two terms for the root mustarise from both the given terms separately-; and if the twoterms of the root cannot thus be brought to agree bothways, such binomial or residual will not be a cube.”

If therefore the expression for the root of a cubic equa-tion should lead to the expression %/(\/\08 + 10) —s/( v/108 — 10); then since,as above^/( ^/108 + 10)is= \S3 + 1;and that, by our theorem, ■/(V 108 — 10) is =\/3—l; there-fore (v/3+1) — (a/3 — l) = 2 is the root of the equationsought.