SECTION III. POWERS AND INVOLUTION. ROOTS. 203. A Power of a quantity is the product of factors each of which is equal to that quantity. A quantity is said to be raised or involved when any power of it is found. 204. Involution is the process of raising a quantity to any given power. 205. Involution is indicated by an exponent, which expresses the name of the power, and shows how many times the quantity is taken as a factor. Thus, 'et a represent any quantity; then, The first power of a is "second 66 66 a = a1; 206. The Square of a quantity is its second power; and the Cube of a quantity is its third power. 207. A Perfect Power is a quantity that can be exactly produced by taking some other quantity a certain number of times as a factor. Thus, x2 — 2xy + y2 is a perfect power, because it is equal to (x − y) (x − y). POWERS OF MONOMIALS. 208. A simple factor may be raised to any power by giving it an exponent which expresses the name or degree of the required power. And if a quantity consists of two or more factors, it is evident that as often as the quantity is repeated, each factor will be repeated. Thus, (ab)2 = ab × ab = aa × bb = a2b2. And in general, if abc.... k represent the product of any number of factors, and n any exponent, we shall have n n (abc . . . . k)" = a"b"c" ...k". That is, The nth power of the product of two or more factors is equal to the product of the nth powers of those factors. 209. If it be required to involve a quantity which is already a power, the exponent of the quantity will be taken as many times as there are units in the exponent of the required power. Thus, (am)2 = am × am (am)3 = am × am × am And in general, am raised to the nth = am+m =a2m; (am)n = amn 3m am+m+m = a3m. power will be That is, If the mth power of a quantity be raised to the nth power, the result will be a power of the quantity expressed by the product of m and n. 210. With respect to signs, it is obvious that if a positive quantity be involved to any power, the result will be positive. But if a negative quantity be involved, the successive powers will be alternately positive and negative; for, it has been shown that the product of an even number of negative factors is positive, and the product of an odd number of negative factors is negative (67). To deduce this law of signs in an experimental way, let it be required to involve a to successive powers. By the principles of multiplication, we shall have the plus sign in the second member being used when n is even, and the minus sign when n is odd. Hence, 1. All powers of a positive quantity are positive. 2. The odd powers of a negative quantity are negative, but the even powers are positive. 211. From the foregoing principles relating to the involution of a monomial, we derive the following RULE.-I. Raise the numeral coefficients to the required power. II. Multiply the exponent of each letter by the exponent of the required power. III. When the quantity involved is negative, give the odd powers the minus sign. 212. If it be required to raise am to the mth power, we shall have (am)m = am×m = am2, an expression which denotes that power of a whose exponent is m2. If we put m = Expressions like the above may frequently occur in algebraic operations. EXAMPLES. Find the value of each of the following expressions: 213. If a fraction be raised to any power, both numerator and denominator will be raised to the same power. Hence, to raise a fraction to any power, we have the following RULE.-Raise both numerator and denominator to the re DISCUSSION OF NEGATIVE INDICES. 214. It has been shown in previous articles that m am xan = am+n, a a' n = am-n, and (am)n = amn, where m and n are positive whole numbers. It remains to be shown that the above relations hold true when one or both of the exponents are negative. And in this investigation it is sufficient to remember that a quantity with a negative exponent is equal to the reciprocal of the same quantity with a positive exponent (88, 2). I. To prove that am × a" = am+n universally, m and n being integers. 1. Suppose one of the exponents to be negative; or let |