We have, therefore, for the multiplication of quantities when the exponents are negative or fractional, the same rule as when they are positive whole numbers, and consequently, the same rule for the formation of powers.

168. Let it be required to divide a by a. We shall have,

Hence, we see that the rule for the division of quantities, with fractional exponents, is the same as though the exponents were positive whole numbers; and consequently we have the same rule for the extraction of roots, as when the exponents are positive whole numbers.

169. We see from the preceding discussion, that operations to be performed upon radicals, require no other rules than those previously established for quantities in which the exponents are entire. These operations are, therefore, reduced to simple oper ations upon fractions, with which we are already familiar.

4. What is the product of

a2+a2b3 + a3¿3 +ab+a3¿+b3, by a+ _ ¿$.

Ans. a3-b2.

5. Divide a3—a2 ̄† _ ab+bt, by at .. ¿ ̄3⁄4.

-

Ans. a2-b.

170. If we have an exponent which is a decimal fraction, as for example, in the expression 10 301 from what has gone be

fore the quantity is equal to (10)1000, or to 1000 000/(10)301, the

value of which it would be impossible to compute, by any process yet given, but which will hereafter be shown to be nearly equal to 2. In like manner, if the exponent is a radical, as √√3, 3/11, &c., we may treat the expression as though the exponents were fractional, since its values may be determined, to any degree of exactness, in decimal terms.

171. A SERIES, in algebra, consists of an infinite number of terms following one another, each of which is derived from one or more of the preceding ones by a fixed law. This law is called the law of the series.

Arithmetical Progression.

172. An ARITHMETICAL PROGRESSION is a series, in which each term is derived from the preceding one by the addition of a constant quantity called the common difference.

If the common difference is positive, each term will be greater than the preceding one, and the progression is said to be increasing.

If the common difference is negative, each term will be less than the preceding one, and the progression is said to be decreasing.

Thus, 1, 3, 5, 7,

&c., is an increasing arithmetical

progression, in which the common difference is 2

and . . . . . 19, 16, 13, 10, 7, 19, 16, 13, 10, 7, . . . is a decreasing arithmetical progression, in which the common difference is 3.

173. When a certain number of terms of an arithmetical progression are considered, the first of these is called the first term of the progression, the last is called the last term of the progression, and both together are called the extremes. All the terms between the extremes are called arithmetical means. An arithmetical progression is often called a progression by differences.

174. Let d represent the common difference of the arithmetical progression,

α b с e.f.g h . k, &c.,

which is written by placing a period between each two of the

terms.

From the definition of a progression, it follows that,

bad, c=b+da+2d, e=c+da+3d;

and, in general, any term of the series, is equal to the first term plus as many times the common difference as there are preseding terms.

Thus, let

be any term, and n the number which marks the Then, the number of preceding terms will be denoted by n 1, and the expression for this general term, will be

place of it.

If d is positive, the progression will be increasing; hence, In an increasing arithmetical progression, any term is equal to the first term, plus the product of the common difference by the number of preceding terms.

If we make n = 1, we have la; that is, there will be but one term.

that is, there will be two terms, and the second term is equal to the first plus the common difference.

3. If a 7 and d5. what is the 9th term?

Ans. 47.

serves to find any term whatever, without determining those which precede it.