arithmetical determination of the root of the degree m n of the number an fm. If we would give a determinate value to the product of the m radical quantities √±a, √±b, by fixing the degree of the radical signs, we must obtain from the equations the several expressions for 1, 1, and combine them in a suitable manner. To conclude, these operations are not often required, except in some very simple cases, of which the following are the principal; 1. v=ax v=b = √ā × √ò (v=1X √1); I suppress the radical sign in the expression v=ax v=bab x-1= 2. 1, and obtain √ ab. 1, because this would lead to I do not here multiply - 1 by the ambiguity mentioned in art. 173; but observing, that the square of the fourth root is simply the square root, we have The results will be thus found to be alternately real and imaginary. Calculus of Fractional Exponents. 175. If we substitute in the place of the radical signs, their corresponding fractional exponents (132), and apply immediately the rules for the exponents, we shall obtain the same results, as those furnished by the methods employed in the calculus of radical quantities. If we transform, for example, Let there be the general example at ba × √√b" c2; the radical expressions here employed may be transformed into we then have, according to the rules for exponents, (25), the same denominator; and to give uniformity to the results, we must do the same with respect to the fractions P S m n ; we obtain, by this means, np ng+mr ms amn b mn cmn; and placing this result under the radical sign, we have √uP bq × √/b*c = √a^p bng+mr cms. 176. The manner of performing division is equally simple, we have for example reducing the fractional exponents to the same denominator, in order to perform the subtraction, which is required, we find It is obvious, that the reduction of fractional exponents to the same denominator, answers here to the reduction of radical expressions to the same degree, and leads to precisely the same results (171). 177. It is also very evident, by the rule given in art. 127, that and by the rule laid down in art. 129, that The calculus of fractional exponents affords one of the most remarkable examples of the utility of signs, when well chosen. The analogy which prevails among exponents, both fractional and entire, renders the rules, that are to be followed with respect to the latter, applicable also to the former; but a particular investigation is necessary in each case, when we use the sign ✔, because it has no connexion with the operation that is indicated. The further we advance in algebra the more fully shall we be convinced of the numerous advantages, which arise from the notation by exponents, introduced by Descartes. General Theory of Equations. 178. EQUATIONS of the first and second degree are, properly speaking, the only ones, which admit of a complete solution; but there are general properties of equations of whatever degree, by which we are able to solve them, when they are numerical, and which lead to many conclusions, of use in the higher parts of algebra. These properties relate to the particular form, which every equation is capable of assuming. An equation in its most general form must contain all the powers of the unknown quantity, from that of the degree of the equation to the first degree, multiplied each by some known quantity, together with one term wholly known. A general equation of the fifth degree, for example, contains all the powers of the unknown quantity, from the first to the fifth; and if there are several terms involving the same power of the unknown quantity, we must suppose them to be united in one; according to the method given for equations of the second degree, art. 108. All the terms of the equation are then to be brought into one member, as in the article above referred to; the other member will necessarily be zero; and when the first term is negative, it is rendered positive by changing the signs of all the terms of the equation. In this way we obtain an expression similar to the following; nx2 + px1 + q x2 + rx2 + s x + t = 0, in which it is to be observed, that the letters n, p, q, r, s, t, may represent negative as well as positive numbers; then dividing the whole by n, in order that the first term may have only unity for its coefficient, and making x2 + Px2 + Qx3 + Rx2 + S x + T = 0. In future, I shall suppose, that equations have always been prepared as above, and shall represent the general equation of any degree whatever by The interval denoted by the points may be filled up, when the exponent n takes a determinate value. Every quantity or expression, whether real or imaginary, which, put in the place of the unknown quantity x in an equation prepared as above, renders the first member equal to zero, and which consequently satisfies the question, is called the root of the proposed equation; but as the inquiry does not at present relate to powers, this acceptation of the term root is more general, than that, in which it has hitherto been used (90, 129). 179. Take a proposition analogous to those given in articles 116 and 159, and one which may be regarded as fundamental. If the root of any equation whatever, n-2 x2 + P x2¬1 + Q x2-2..... + T x + U = 0, be represented by a, the first member of this equation may be exactly divided by x -a. Indeed, since a is one value of x, we have, necessarily, an + Pan-1 + Q a2-2 ..... + Ta + U = 0, so that the equation proposed is precisely the same as are each divisible by xa (158), it is evident, that the first member of the proposed equation is made up of terms, all of which are divisible by this quantity, and may consequently be divided by x-a, as the enunciation of the proposition requires.* *D'Alembert has proved the same proposition in the following manner. If we conceive the first member of the proposed equation to be divided by x a, and the operation continued until all the terms involving x are exhausted, the remainder, if there be any, cannot contain x. If we represent this remainder by R, and the quotient to which we arrive by Q, we have necessarily x2 + P xn−1 ..... + &c. = Q(x − a) + R. Now if we substitute a in the place of x, the first member is reduced to nothing, since a is the value of x; the term Q (x a) is also nothing, because the factor x a becomes zero; we must, therefore, have R=0, and it is so, independently of the substitution of a; for, as this remainder does not contain x, the substitution cannot take place, and it still preserves the value it had before. Hence it follows, that in every case, R= 0, and that, consequently, +P+Q-2, &c. is exactly divisible by x-a. |