sequently, the 6th root of a, admits of six values. If we make

We may then conclude from analogy, that in every equation of the form xm - a = 0, or rm - pm = 0, x is susceptible of m different values; that is, the mth root of a number admits of m different algebraic values.

231. If, in the preceding equations, and the results corresponding to them, we suppose, as a particular case, a = 1, whence p = 1, we shall obtain the second, third, fourth, &c. roots of unity. Thus + 1 and 1 are the two square roots of unity, because the equation x2 In like manner,

1 = 0, gives x = ± 1.

are the three cube roots of unity, or the roots of x3- 1 = 0; and

are the four fourth roots of unity, or the roots of x4 - 1 = 0.

232. It results from the preceding analysis, that the rules for the calculus of radicals, which are exact when applied to absolute numbers, are susceptible of some modifications, when applied to expressions or symbols which are purely algebraic; these modifications are more particularly necessary when applied to imaginary expressions, and are a consequence of what has been said in Art. 230.

For example, the product of

a by ✓ by the rule of Art. 228, would be

a X

a,

- a = + a2.

Now, a2 is equal to a (Art. 139); there is, then, apparently, an uncertainty as to the sign with which a should be affected. Nevertheless, the true answer is -a; for, in order to square m, it is only necessary to suppress the radical; but

Again, let it be required to form the product

-ax-b.

By the rule of Art. 228, we shall have

√-ax√-b=

+ ab.

Now, √ab = p (Art. 230), p being the arithmetical value

of the square root of ab; but the true result is

so long as both the radicals

with the sign +.

- a

and

por

√ab,

- b are affected

For, √-a=√a.√ - 1; and √ - b = √b. - 1,

hence,

√-ax√- b = √a.√ - 1 x √bx√ - 1 = √ab (-1)2

= √ab x − 1 = −√ab.

By similar methods we find the different powers of be as follows:

1.

2.

3.

√-1x-1= ( − 1)2 = − 1.

(-1)3 = (-1)2. √ − 1 = - -1.

2

=-1x-1=

(-1) = (-1)2. (√ – 1)2 = − 1 x − 1 = + 1.

1 to

Again, let it be proposed to determine the product of a by the which, from the rule, would be + ab, and con

sequently, would give the four values (Art. 231),

+√ab, Vab, +√ab. √ -1, -Vab.-1.

To determine the true product, observe that

We will apply the preceding calculus to the verification of the

considered as a root of the equation x3 1 = 0; that is, as one

of the cube roots of 1 (Art. 230).

(a + b)3 = a3 + 3a2b + 3ab2 +63,

From the formula,

we have

=

(-1)3 + 3 (-1)2. – 3 + 3 (-1). (-3)2 + (-3)3

manner. It should be remarked, that either of the imaginary roots is the square of the other; a fact which may be easily verified.

Theory of Exponents.

233. In extracting the nth root of a quantity am, we have seen that when m is a multiple of n, we should divide the exponent m by n the index of the root. When m is not divisible by n, the operation of extracting the root is indicated by indicating the division of the two exponents. Thus,

m

"Vam = an,

a notation tounded on the rule for the exponents, in the extraction of the roots of monomials. In such expressions, the numerator indicates the power to which the quantity is to be raised, and the denominator, the root to be extracted.

Therefore,

3

Va3 = a3;

and Va =

7 a4.

If it is required to divide am by a", in which m and n are positive whole numbers, we know that the exponent of the divisor should be subtracted from the exponent of the dividend, and we have

If m>n, the division will be exact; but when m <n, the division cannot be effected, but still we subtract, in the algebraic sense, the exponent of the divisor from that of the dividend. Let p be the arithmetical difference between n and m; then will

Therefore, the expression a P is the symbol of a division which has not been performed; and its true value is the quotient represented by unity, divided by a, affected with the exponent p, taken positively. Thus,

1

1

ap

a-P

Since, a-r = -; and = p", we conclude that,

Any factor may be transferred from the numerator to the denominator, or from the denominator to the numerator, by changing the sign of its exponent.

1

If it is required to extract the nth root of we have

am

The notation of fractional exponents, whether positive or nega

tive, has the advantage of giving an entire form to all expres

sions whose roots or powers are to be indicated.

From the conventional expressions founded on the preceding rules, we have

We may therefore substitute the second value in each expres

sion, for the first, or reciprocally.

As ar is called a to the p power, when p is a positive whole number, so, by analogy,

m

a to the

n

power, in which algebraists have generalized the word power. It would, perhaps, be more accurate to say, a, ex

m

ponent a, exponent - p, and a, exponent

n

word power only when we wish to designate the product of a number multiplied by itself two or more times.

reducing to a common index (Art. 226), and then multiplying,

Therefore, in order to multiply two monomials affected with any exponents whatever, follow the rule given in Art. 41, for quantities affected with entire exponents.