Division of Quantities affected with any Exponents.

235. To divide one monomial by another when both are affected with any exponent whatever, divide the co-efficient of the dividend by that of the divisor, for a new co-efficient: subtract the exponent of each letter in the divisor from the exponent of the same letter in the dividend, and then annex each letter with its new exponent.

For, the exponent of each letter in the quotient must be such, that, added to the exponent of the same letter in the divisor, the sum shall be equal to the exponent of the letter in the dividend; hence, the exponent of any letter in the quotient, is equal to the difference between the exponent of that letter in the dividend and in the divisor.

236. To form the mth power of a monomial, affected with any exponents whatever, raise the co-efficient to the mth power, and for the exponents, observe the rule given in Art. 220, viz, multiply the exponent of each letter by the exponent m of the power.

For, to raise a quantity to the mth power, is the same thing as to multiply it by itself m1 times; therefore, by the rule for multiplication, the exponent of each letter must be added to itself - 1 times, or multiplied by m.

m

237. To extract the nth root of a monomial, extract the nth root of the co-efficient, and for the new exponents, follow the rule given in Art. 220, viz., divide the exponent of each letter by the index of the root."

For, the exponent of each letter in the result should be such, that when multiplied by n, the index of the root to be extracted, the product will be the exponent with which the letter is affected in the proposed monomial; therefore, the exponents in the result must be respectively equal to the quotients arising from the division of the exponents in the proposed monomials, by n, the index of the root. Thus,

The rules for fractional and negative exponents have been easily deduced from the rule for multiplication; but we may give a direct demonstration of them, by going back to the origin of quantities affected with such exponents.

We will demonstrate implicitly, the two preceding rules.

By going back to the origin of these notations, we find that

REMARK I.-The advantage derived from the use of fractional exponents consists principally in this:-The operations performed upon expressions of this kind require no other rules than those established for the calculus of quantities affected with entire exponents. This calculus is thus reduced to simple operations upon fractions, with which we are already familiar.

3

REMARK II. In the resolution of certain questions, we shall be led to consider quantities affected with incommensurable exponents. Now, it would seem that the rules just established for commensurable exponents, ought to be demonstrated for the case in which they are incommensurable. But let us observe, that the value of an incommensurable, such as √3, VII, may be determined approximatively as near as we please, so that we can always conceive the incommensurable to be replaced by an exact fraction, which only differs from it by a quantity less than any given quantity; and we apply the rules to the symbol which designates the incommensurable, after substituting the fraction which represents it approximatively.

4. What is the product of

až + a2b§ + a3b3 + ab + a3b3 + b§,

b3, by

5. Divide

a} — a2b ̃} _ a3b + b},

by

Method of Indeterminate Co-efficients.

238. The binomial theorem demonstrated in Art. 203, explains the method of developing into a series any expression of the form (a+b)m, in which m is a whole and positive number.

Algebraists have invented another method of developing algebraic expressions into series, called the method by indeterminate co-efficients. This method is more extensive in its applications, can be applied to algebraic expressions of any nature whatever, and indeed, the general case of the binomial theorem may be demonstrated by it:

Before considering this method, it will be necessary to explain what is meant by the term function.

In this equation, a, b, and c, mutually depend on each other for their values. For,

The quantity a is said to be a function of b and c, b a function of a and c, and c a function of a and b. And generally, when one quantity depends for its value on one or more quantities, it is said to be function of each and all the quantities on which it depends.

239. If we have an equation of the form,

A+ Bx+Cx2 + Dx3 + Ex2 + &c. = 0;

it is required to find the values of the co-efficients A, B, C, D, E, &c., under the following suppositions :

1st. That no one of the co-efficients is a function of x.

2d. That the series shall be equal to zero, whatever be the number of its terms; and

3d. That it shall be equal to zero, whatever value may be attributed to x.

Now, since the co-efficients are independent of x, their values cannot be affected by any supposition made on the value of x: hence, if they be determined for one value of x, they will be known for all values whatever.

Let us now make

x = 0, which gives

Bx + Сx2 + Dx3 + Ex1 + &c. = 0;

and consequently,

A = 0.

If we divide by x, we have

B+C+Dax2 + Ex3 + &c. =

and by again making x 0, we have

=

Cx+Dx2+ Ex3 + &c. = 0;

0,

and by continuing the process we may prove that, each co-efficient must be separately equal to zero.

It should be observed, that A may be considered the co-efficient of xo.

240. The principle demonstrated above, may be enunciated under another form. If we have an equation of the form

...

a + bx + cx2 + dx3 + = a + b'x + c2x2 + d2x2 + ... which is satisfied for any value whatever attributed to x, the coefficients of the terms involving the same powers of x in the two members, are respectively equal. For, by transposing all the terms into the first member, the equation will take the form

Every equation in which the terms are arranged with reference to a certain letter, and which is satisfied for any value which may be attributed to that letter, is called an identical equation, in order to distinguish it from a common equation, that is, an equation which can only be satisfied by particular values of the unknown quantity.