in which the square l' of the binomial m+n, produces, when developed, the terms m2 + 2 m n + n3. Now, after we have arranged the proposed quantity, the first term will evidently be the square of the first term of the root, and the second will contain double the product of the first term of the root by the second of this root; we shall then obtain this last by dividing the second term of the proposed quantity by double the root of the first. Knowing then the two first terms of the root sought, we complete the square of these two terms, represented here by 1; subtracting this square from the proposed quantity, we have for a remainder 21p+p2, a quantity, which contains double the product of l, or of the first binomial m+n, by the remainder of the root, plus the square of this remainder. It is evident therefore that we must proceed with this binomial as we have done with the first term m of the root. Let there be, for example, the quantity 64 a2 b c + 25 a2 b2 — 40 a3 b + 16 aa + 64 b3 ca 80 a b c ; we arrange it with reference to the letter a, and make the same disposition of the several parts of the operation as in the above example. 16a40a3b+25a2 b3 — 80ab3c + 64b9c2 -4a2 5ab+8bc +64a2bc. -80ab2c + 64b2 c We extract the square root of the first term 16 aa, and obtain 4 a2 for the first term of the root sought, the square of which is to be subtracted from the proposed quantity. We double the first term of the root, and write the result 8 a2 under the root; dividing by this the term 40 a3 b, which be- a3 gins the first remainder, we have 5 a b for the second term of the root; this is to be placed by the side of 8 a2, we then multiply the whole by this second term, and subtract the result from the remainder, upon which we are employed. Thus we have subtracted from the proposed quantity the square of the binomial 4 a 5 ab; the second remainder can contain only double the product of this binomial, by the third term of the root, together with the square of this term; we take then double the quantity 4 a-5 a b, or 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 4a25ab8bc. 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 perceiv ed 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 5th 1 32 81 256 625 243 1024 3125 7776 16807 32768 59049 6th 64 729 409615625 46656 117649 262144 531441 7th 1128 2187 16384 78125 279936 823543 2097152 4782969 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 18 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 a bo c3; the operation may be thus represented, a2 b3 cx a2 b3 cx a2 b3 c = a2.3b3.3 c1.3. 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 c', is 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. 19 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 as b9 c3, by dividing the exponents 6, 9 and 3 by S, 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 3 ab2c5. 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 fuctors indicated by the sign. 96 a b c1125 a5 b5 c10 × 3 b2 c. As the first factor 25 a5 b5 c10, has for its fifth root the quantity 2 a b c2, 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+ √—ab”, 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 |