Thus we have subtracted from the proposed quantity the square of the binomial 4 a2 -5 ab; the second remainder can contain only double the product of this binomial, by the third term of the root, together with the double the quantity 4 a - square of this term; we take then 5 a b, or 8 a 10 ab, - which is written under 8 a 5 a b, and constitutes the divisor to be used with the second remainder; the first term of the quotient, which is 8 b c, is the third of the root. This term we write by the side of 8 a2 - 10 ab, and multiply the whole expression by it; the product being subtracted from the remainder under consideration, nothing is left; the quantity proposed therefore is the square of 4 a5ab8bc. The above operation, which is perfectly analogous to that, which has been already applied to numbers, may be extended to any length we please. Of the formation of powers and the extraction of their roots. 126. The arithmetical operation, upon which the resolution of equations of the second degree depends, and by which we ascend from the square of a quantity to the quantity, from which it is derived, or to the square root, is only a particular case of a more general problem, namely, to find a number, any power of which is known. The investigation of this problem leads to a result, that is still termed a root, the different kinds being called degrees, but the process is to be understood only by a careful examination of the steps by which a power is obtained, one operation being the reverse of the other, as we observe with respect to division and multiplication, with which it will soon be perceived that this subject has other relations. It is by multiplication, that we arrive at the powers of entire numbers (24), and it is evident, that those of fractions also are formed by raising the numerator and denominator to the power proposed (96). So also the root of a fraction, of whatever degree, is obtained by taking the corresponding root of the numerator and that of the denominator. As algebraic symbols are of great use in expressing every thing, which relates to the composition and decomposition of quantities, I shall first consider how the powers of algebraic expressions are formed, those of numbers being easily found by the methods that have already been given (24.)' Table of the first seven powers of numbers from 1 to 9. 4th 1 16 1296 2401 4096 6561 81 256 625 5th 1 32 243 1024) 3125 32768 59049 6th 64 729 4096 15625 46656 117649 262144) 531441 7th 1128 2187 16384 78125|279936|823543 20971524782969 This table is intended particularly to show with what rapidity the higher powers of numbers increase, a circumstance that will be found to be of great importance hereafter; we see, for instance, that the seventh power of 2 is 128, and that of 9 amounts to 4782969. It will hence be readily perceived, that the powers of fractions, properly so called, decrease very rapidily, since the powers of the denominator become greater and greater in comparison with those of the numerator. The seventh power of, for example, is, and that of is only 148 1 4782969 127. It is evident from what has been said, that in a product each letter has for an exponent the sum of the exponents of its several factors (26), that the power of a simple quantity is obtained by multiplying the exponent of each factor by the exponent of this power. The third power of a2 b3 c, for example, is found by multiplying the exponents 2, 3, and 1, of the letters a, b and c, by 3, the exponent of the power required; we have then as b9c3; the operation may be thus represented, a2 b3 cx a2 b3 cx a2 b3 c=a2.3b3.3 cl.3. If the proposed quantity have a numerical coefficient, this coefficient must also be raised to the same power; thus the fourth power of 3 a b2 c3, is 81 ab eo, 128. With respect to the signs, with which simple quantities may be affected, it must be observed, that every power, the expoent of which is an even number, has the sign +, and every power the exponent of which is an odd number, has the same sign as the quantity from which it is formed. In fact, powers of an even degree arise from the multiplication of an even number of factors; and the signs, combined two and two in the multiplication, always give the sign + in the product (31). On the contrary, if the number of factors is uneven, the product will have the sign, when the factors have this sign, since this product will arise from that of an even number of factors, multiplied by a negative factor. 129. In order to ascend from the power of a quantity, to the root from which it is derived, we have only to reverse the rules given above, that is, to divide the exponent of each letter by that, which marks the degree of the root required. Thus we find the cube root, or the root of the third degree, of the expression a bo c3, by dividing the exponents 6, 9 and 3 by 3, which gives a2 b3 c. When the proposed expression has a numerical coefficient, its root must be taken for the coefficient of the literal quantity, obtained by the preceding rule. If it were required, for example, to find the fourth root of 81a4 b3 c2o, we see by referring to table art. 126, that 81 is the fourth power of 3; then dividing the exponent of each of the letters by 4, we obtain for the result Sab2 cs. When the root of the numerical coefficient cannot be found by the table inserted above, it must be extracted by the methods to be given hereafter. 130. It is evident, that the roots of the literal part of simple quantities can be extracted, only when each of the exponents is divisible by that of the root; in the contrary case, we can only indicate the arithmetical operation, which is to be performed, whenever numbers are substituted in the place of the letters. We use for this purpose the sign; but to designate the degree of the root, we place the exponent as in the following expressions, the first of which represents the cube root, or the root of the third degree of a, and the second the fifth root of a2. We may often simplify radical expressions of any degree whatever, by observing, according to art. 127. that any power of a product is made up of the product of the same power of each of the factors, and that consequently, any root of a product is made up of the product of the roots of the same degree of the several factors. It follows from this last principle, that, if the quantity placed under the radical sign have factors, which are exact powers of the degree denoted by this sign, the roots of these factors may be taken separately, and their product multiplied by the root of the other factors indicated by the sign. 96 a5 b7c11 = 28 a5 b5 c10 × 3b2 c. As the first factor 25 a5 b5 c1o, has for its fifth root the quantity 2 a b c, the expression becomes 131. As every even power has the sign + (128), a quantity, affected with the sign, cannot be a power of a degree denoted by an even number, and it can have no root of this degree. It follows from this, that every radical expression of a degree which is denoted by an even number, and which involves a negative quantity, is imaginary, thus 6 —a, √=a2, b+√=ab2, are imaginary expressions. We cannot therefore, either exactly or by approximation, assign for a degree, the exponent of which is an even number, any roots but those of positive quantities, and these roots may be affected indifferently with the sign + or —, because in either case, they will equally reproduce the proposed quantity with the sign+, and we do not know to which class they belong. The same cannot be said of degrees expressed by an odd num ber, for here the powers have the same sign as their roots (128); and we must give to the roots of these degrees the sign, with which the power is affected; and no imaginary expressions occur. 132. It is proper to observe, that the application of the rule given in art. 129, for the extraction of the roots of simple quantities, by means of the exponent of their factors, leads to a more convenient method of indicating roots, which cannot be obtained algebraically, than by the sign √. If it were required, for example, to find the third root of a3, it is necessary, according to the rule given above, to divide the exponent 5 by 3; but as we cannot perform the division, we have for the quotient the fractional number §; and this form of the exponent indicates, that the extraction of the root is not possible in the actual state of the quantity proposed. We may therefore consider the two expressions The second however has this advantage over the first, that it leads directly to a more simple form, which the quantity 3 ✔a is capable of assuming; for if we take the whole number contained in the fraction, we have 1+ as an equivalent exponent; consequently, a = a1+3= = a1× a3 (25); from which it is evident, that the quantity as is composed of two factors, the first of which is rational, and the other becomes 3 The same result indeed may be obtained from the quantity under the form, by the rule given in art. 130, but the fractional exponent suggests it immediately. We shall have occasion to notice in other operations the advantages of fractional exponents. We will merely observe for the present, that as the division of exponents, when it can be performed, answers to the extraction of roots, the indication of this division under the form of a fraction is to be regarded as the symbol of the same operation; whence, |