BOOK III CHAPTER I INVOLUTION 260. If a quantity is repeatedly multiplied by itself, it is said to be raised to a power, or involved, and the power to which it is raised is expressed by the number of times the quantity has been used as a factor in the multiplication. The operation is called Involution. Thus, as has been stated, a Xa or a2 is called the second power of a; a × a × a or a3 is called the third power of a; and so on. In 285, 86, 89, VIII, some examples in involution have been given, but it is now desired to give additional rules more concisely stated and of more general character. The theory of involution, however, involving fractional and negative exponents, will not be discussed now, but later in a chapter on the theory of exponents. 261. Index Law for Involution. it has been proved, in 285, 4, that In case n is a positive integer (am)n = amn. Hence, any required power of a given power of a number is found by multiplying the exponent of the given power by the exponent of the required power. 262. The Law of Signs. If the quantity to be raised to a given power has a negative sign, the sign of the even powers will be positive, and the sign of the odd powers will be negative. (— a)3 = (— a) ( − a) ( − a) = (+ a)2 ( − a) = — a3 (a) = (-a)2 (— a)2 = ( + a3) (+ a3) = + a1 (— a)2n = [( − a)2]" = ( a2)" = + a2n [261] (-a)2n+1 = (— a) (— a)2n — (— a) (+ a2n) — — a2n+1 ̧ — : = Here is any positive integer. These results show that, when the exponent is even the result of the involution has the sign, and when the exponent is odd, the result has the sign. 263. The Positive Integral Power of a Positive Quantity. It has been proved that 1. (an)m = amn [261] In case n is a positive integer, it also follows from 85, 5, that 3. (abe)" (abc) (abe) (abc)" = = (abc) (abe) ... to the product of n factors (abc) (a · a to n factors) · (b ⋅ b · . . . to n factors) (—am)n = ±an, where the positive or negative sign is to be prefixed according as n is even or odd. Or, since which is or = (-1)" amn, [2] according as n is even or odd. These five observa tions give the following rule: A quantity is raised to any power by multiplying the exponent of every factor in the quantity by the exponent of that power, and prefixing the proper sign, determined by the preceding rule. 264. The Positive Integral Power of a Fraction. By definition, when is a rational fraction, 265. Powers of Binomials. It has already been proved in 189 that 1. (a + b)2 = a2+2ab+b2, the second power of (a + b) 2. (a+b)3 a3 + 3 ab+3ab2+63, third power of (a + b) 3. (a+b) a1+4a3b+6a2b+4ab3+b', fourth power of (a+b): Similarly, the second, third, and fourth powers of (a — b) are: 4. (a - b)2 = a2 − 2 ab + b2 5. (a —b)3 = a3 — 3 a2b + 3 ab2 — b3 6. (a — b)1 = a1 — 4a3b + 6 a2b2 — 4 ab3 + b1 That is, wherever the odd power of b occurs, the negative sign is prefixed. Later the theorem called the Binomial Theorem will be proved which provides a method for finding any positive integral power of the binomials a + b or a b without multiplication. This theorem has been stated in 289, VIII. It may be expressed for the exponent n in a formula as follows: These formulae have n + 1 terms in case n is a positive integer, but have an infinite number if n be negative or fractional. 266. These rules for the formation of a power of a binomial hold in case the terms of the binomial have coefficients or exponents. 1. Find the third power of 2 x2 — 3 y3. Since (a - b)3 = a3 3 ab+3ab2 — b3, by putting 22 for a and 3 y3 for b, it follows that = (2x2-3 y3)3 (2 x2)3 — 3 (2 x2)2 (3 y3)+3(2 x2) (3 y3)2 — (3 y3)3 =8x36x1y3 +54x2y - 27 y'. 2. Find the fifth power of 2 Since by putting (a — b)5 = a3 — 5 a1b + 10 a3b2 — 10 a2b3 + 5 ab1 — b3, for a and y≈ for b, the result is (x2)5 — 5 (x2)1 (} y2z) + 10 (x2)3 (1 y°z)2 - 10 (x2)2 (1 y°z)3 + 5 (x2) ( } y2z)1 — (!y2z)5 =x10-} x8y2 z + { x$y$z2 — {x1y® z3 + $6x278 z1 — 32y1025. 267. It is evident that the mth power of a" is the same thing as the nth power of a", namely, amn; that is, the same result is arrived at by different processes of involution. For example, the 6th power of a+b may be found by repeated multiplication by (a+b); or *It will be shown later that the law of formation of these formulae holds when n is a a negative integer or a positive or negative fraction when −1<<+1. the cube of a+b may first be found and then the square of the result, since the square of (a + b)3 is (a + b)"; or the square of (a+b) may first be found and then the cube of (a + b)2, which is (a+b). 268. Powers of Expressions of more than Two Terms.-It has already been shown (289, VIII) that (a + b + c)2 = a2 + b2 + c2 + 2 ab + 2 ac + 2 be, (a+b+c+d)2=a2+b2+c2+d2+2ab+2 ac+2ad+2bc+2bd+2cd; and hence is obtained the following rule, which holds good in the preceding examples and others similar to them: The square of any polynomial consists of the square of each term, together with twice the product of every pair of terms. These results may be written in another form: (a + b + c)2 = a2 + 2 a (b + c) + b2 + 2 be + c2 (a+b+c+d)2 = a2 + 2 a (b+c+d) +b2+ 2b (c+d) +c2+ 2cd+d2. The following rule holds good in these and similar examples: The square of any polynomial consists of the square of each term plus twice the product of each term by the sum of all the terms which follow it. A general proof of these formulae for general cases can be deduced by the process of mathematical induction, which will be Thus, it may be proved that: explained later. 269. The following additional examples illustrate the first of the rules in the preceding article. (a + b — c)2 = a2 + b2 + c2 + 2 ab 2 ac 2 bc. -- (12x+3)2 = 1 + 4x2+9xx1 — 4x + 6 x2 - 12 x3 14x10x12x+9x1. (1-2x+3x-4)=1+4x+9x+16x-4x+6x-8x -12x+16 x 24 x 1-4x+10x20 x3+25 x1-24x+16x6. |