POWER, in arithmetic, and in the algebra of real numbers, the product obtained by multi plying a given number by itself a specified number of times. If x is the given number, and n is the number of times that x is taken as .a factor in forming the product, then the product that is finally obtained is xXxXxX . . . XxXxXz (n factors altogether) This is called the ((nth power" of the number .r, and is represented by the symbol xn. As suming for the moment that n (which is called the °exponent') of the power) is positive and in tegral (since this is the only case in which the foregoing definition is applicable), we easily obtain the following general laws which the powers of numbers must fulfil: en-1-01 . (en) n = ; ( n xer. To extend the conception of a so as to permit of the use of fractional and negative ex ponents, we may assume that the foregoing laws hold true of all real exponents, and from them we may seek the interpretations that must be !even to an expression of the form when n is fractional or negative. First, if we make nt=0 in the equation xm.x% we have xe.xle=.r*; and 'hence we conclude that the symbol x° must 'be interpreted as representing unity in all cases. Again. if we make is nt, we have, from the first of the general relations given above, xm.x— = — = x° = 1
and hence we must interpret the expression x-61 as signifyino' the reciprocal of Xn. We may ascertain the significance of a fractional ex ponent by means of the second general relation, above. Thus in the identity let 1 us put m --=1. Then we have (--) * xn x' =x. Hence 1 — must be interpreted as rep resenting one of those quantities which, when raised to the nth power, will yield x itself. That is, we must interpret ) as equivalent to In general, this will be a many valued func tion. If x is real, V x is often interpreted as the positive real number which when raised to the nth power gives x. This is only one of the 1 vahles xm of — . In a similar manner it is easily shown that the must be interpreted as equivalent to The significance of a power. when, the ex ponent is irrational or imaginary, is considered in the theory of functions, but these generaliza tions are too difficult for sound •treatment in the present article. For this aspect of the sub ject, and for the discussion of the development of functions of a variable in a series whose terms are powers of that variable, consult Harkness and Morley, 'Introduction to the Theory of Analytic Functions> (London 1898).