TRACT 10. RULE FOR EXTRACTING ROOTS.

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very accurate, and so simple and easy to use and to keep inmind, that nothing more so can be desired or hoped for; andfurther, that instead of searching out rules severally for eachroot, one after another, our investigation is at once for anyindefinite possible root, by whatever quantity the index isexpressed, whether fractional, or irrational, or simple, orcompound.

2. In every theorem, or rule, here investigated,

N denotes the given number, whose root is sought,n the index of that root,

a its nearest rational root, or a” the nearest rational powerto n, whether greater or less,

x the remaining part of the root sought, which may beeither positive or negative, namely, positive when n isgreater than a ”, otherwise negative. Hence then, thegiven number

N is = (a + x) n , and the required root n” = a + x.

3. Now, for the first rule, expand the quantity (a + .r)” bythe binomial theorem, so shall we have

N =z (a -f. x) n = a” + na"~ 1 x + w - ^ ^ -a n ~*x z + & c »

Subtract a " from both sides, so shall

„ „ n — 1

n - a* = n a'-'x -f n . ——-a n - % x 1 + &c.

N —ft" N — ft"

or ~~~ x a

Divide by na'~ *, so shall

na "

— x ft = X +

n — 1 x 1

-+-

■ l n — 2 x ia L

+ &c.

/v*+ 4 U X it-

Here, on account of the smallness of the quantity x in respect°f all the terms of this series, after the first term, will beVer y small, and inay therefore be neglected without much

error, which o-ives -—-ft for a near value of .r, being only ato na”

small matter too great. And consequently

**+*• = i s nearly = n" the root sought. And

n a”

this may be accounted the first theorem.