THE EXPONENTIAL THEOREM.

[310. To expand a* in a series of powers of x.

B, C, &c. containing powers of a-1 only;

Since a* is clearly independent of n, n may be any value

(A+Bn+ ...)2 + ...

.

2

whatever; let, then, n = 0;

COR. If Є be that value of a which makes A equal to 1, then

THE MULTINOMIAL THEOREM.

[311. The Multinomial Theorem is a rule or formula for expanding any power of a quantity which consists of more than

two terms.

The expansion of a multinomial may frequently be effected by the Binomial Theorem, as is done for a trinomial in Art. 307; for (a+b+c+d+ &c.)" may be expanded as a binomial by con

sidering any number of terms as one term, and the remainder as another term. But a more general method is as follows:

312. To expand (a+b+c+d+&c.)", when m is a positive integer.

сх

(a+b+c+d+&c.)x ax. bx. ccx. cdx. &c.

= 6

and if e = 2.7182818, expanding by the Exponential Theorem,

Now, as this operation merely exhibits the same quantity expanded in two different ways by the same theorem, the corresponding terms, that is, the terms involving the same powers of a will be equal to each other; therefore equating the coefficient of am on the one side with the coefficient of am on the other, and observing that each separate term on this side of the equation which involves a will be the product of as many terms as there are series to be multiplied, one of which is taken out of each series, and will therefore be of the form.

* Σ stands for the expression "the sum of all the quantities of the form of."

and by giving p, q, r, &c. all the positive integral values which the condition p + q +r+ &c. = m admits of, the several terms of the expansion will be obtained.

COR. 1. If q+r+ s + &c. π, then p = M — T, and the general term becomes

the general term may be obtained from the former by writing ɑ ̧x, ɑ2x2, ɑ ̧x3, &c. in place of b, c, d, &c. respectively, by which it becomes

m (m − 1)... (m − π+1) 1.2.3...q.1.2.3...r.&c.

ao ̄a11a2". &c. .xl+2r+3s+&c. ;

and all the terms of the expansion may be found as before, by giving q, r, s, &c. all possible integral and positive values which the condition q + r + s + &c. =π admits of.

OBS. The proof here given of the Multinomial Theorem extends only to the case of positive integral indices, for by the Exponential Theorem m cannot be any thing but a positive integer. But if the Multinomial be deduced from the Binomial Theorem, then since the latter is proved for fractional and negative indices, the former is also proved to hold for such indices. Ex. Required the term in the expansion of (a - b−c)7 which involves a2b3 c2.

EVOLUTION OF SURDS.

[A practical method of finding the square root of a binomial surd was given in Art. 177; the following is the one more usually adopted.

313. To extract the square root of a quantity which is under the form a +√b.

+ √y=√a + √b,

Assume √x+y=

then x+y+2xy=a+√b;

.. x + y = a

and 2√xy=

(Art. 174):

from these two equations we find x and y, thus:

From this conclusion it appears that the square root of

a + √b can only be expressed by a binomial, one or both of whose terms are quadratic surds, when ab is a perfect

square.

If the proposed surd be of the form a −√b, then we √x-√y=√ a−√b, and proceed as before.

assume

314. It must be observed that this method applies only to cases in which one of the terms of the binomial is a quadratic surd, and the other rational.

If, however, a binomial is proposed which can be put under the form ac + √be, or √e (a+b), its square √ a2c+√bc,

root may be found, by finding the square root of a +√, and multiplying the result by c.