128. With respect to the signs, with which simple quantities may be affected, it must be observed, that every power, the exponent 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

3

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 31 a4 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 a b2 c5.

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 a3.

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.

Let there be, for example,

10

As the first factor, 25 a5 b5 c1o, has for its fifth root the quan tity 2 a b c2, the expression becomes

√96 a5 b c112 abc2 3 b2 ce

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

8

√—a, √—aa, b + √—ab

are imaginary expressions.

- ab1,

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 a5, 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

3

leads directly to a more simple form, which the quantity as is capable of assuming; for if we take the whole number contained in the fraction, we have 1+ as an equivalent exponent; consequently,

from which it is evident, that the quantity as is composed of two

3

factors, the first of which is rational, and the other becomes a2. The same result, indeed, may be obtained from the quantity

3

under the form as, 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 expo

nents.

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,

We have rules then, which result from the assumed manner of expressing powers, which lead to particular symbols, as in art. 37, we arrived at the expression ao = 1.

133. It may be observed here, that as we divide one power by another, by subtracting the exponent of the latter from that of the former (36), fractions of a particular description may readily be reduced to new forms.

By applying the rule above referred to, we have

but if the exponent n of the denominator exceed the exponent m of the numerator, the exponent of the letter a in the second member will be negative.

If, for example, m = 2, n = 3, we have

but by another method of simplifying the fraction

it equal to 1, the expressions

a

In general, we obtain by the rule for the exponents,

In fact, the sign, which precedes the exponent n, being taken in the sense defined in art. 62, shows that the exponent in question arises from a fraction, the denominator of which contains the factor a, n times more than the numerator, which frac

1

tion is indeed ; we may, therefore, in any case which occurs,

an

substitute one of these expressions for the other.

The quantity

lent to

a2 b5

c2 d3'

for example, being considered as equiva

a2 b5 x X

C3 d3'

may be reduced to the following form,

a2 b5 c-2 d-3;

that is, we may transfer to the numerator all the factors of the denominator, by giving to their exponents the sign

Reciprocally, when a quantity contains factors, which have nega tive exponents, we may convert them into a denominator, observing merely to give to their exponents the sign+; thus the quantity a2 b3 c-2d-3,

becomes

a2 b5 c2 d3°

Of the Formation of the Powers of Compound Quantities.

134. We shall begin this section by observing, that the powers of compound quantities are denoted by including these quantities in a parenthesis, to which is annexed the exponent of the power. The expression

(4 a3 -2ab5b2) 3,

for example, denotes the third power of the quantity,

This power may also be expressed thus,

4a22ab+ 5633.

135. Binomials next to simple quantities are the least complicated, yet if we undertake to form powers of these by successive multiplications, we in this way arrive only at particular results, as in art. 34, we obtained the second and third power; thus

It is not easy from this table to fix upon the law, which determines the value of the numerical coefficients. But by considering how the terms are multiplied into each other, we perceive, that the coefficients have their origin in reductions depending on the equality of the factors, which form a power. This is ren dered very evident by an arrangement, which prevents these reductions taking place.

It is sufficient for this purpose to give to the several binomials