CHAPTER II EVOLUTION DEFINITION AND PRINCIPLES 271. Evolution, or the extraction of any root of a given quantity, is the inverse of the operation of raising a certain quantity to a power which will produce the given quantity. The extraction of the 7th root of a quantity undoes the act of raising the 7th root of that quantity to the 7th power. Thus, by definition, 272. Definition of a Root.-1. The definition of the square root of a number has already been learned (? 94). Thus it is known that 2. The meaning of the cube root of a number has also been learned (? 97). Thus it is known that 3 3 Va a and (va)3= a. Since (a + b)3 = a3 + 3a2b + 3ab2 + b3, then 3 3√ (a3 +3a2b+3ab2 + b3) =3v (a + b)3 = a + b. 3. It follows from the definition of evolution, 271, that the nth root of a number is one of the n equal factors of the number. Thus +3 or 3 is one of the two equal factors of 9, and (a + b) is one of the three equal factors of a3 +3a2b+3 ab2 + b3. 273. The radical sign, V, is used to denote the square root, and is placed before the number whose root is desired (94). The radicand is the number or expression whose root is desired. The index of a root is a number which indicates what root of the radicand is to be found, and is written above the radical sign. Thus, the square root of 9 is written 21/9 or 1/9= 3; the fourth root of 16 is written *1/16 = 2; and the nth root of a is written "Va. Here the indices of the roots are respectively 2, 4, and n. 274. A parenthesis, or vinculum, is often used to express the root of a quantity consisting of more than one term. Thus 16+25 means the sum of 1/16 and 25, while 16+25 means the square root 3 of the sum of 16 and 25. Moreover, y3 and the cube root of a3, while 3 3 product x3y3. 3 3. y3 means the product of 3 means the cube root of the Parentheses are sometimes used instead of the vinculum in connection with the radical sign. Thus, the same result may be expressed by 16+ 25 or 1 (16+25). 275. Like and Unlike Roots.-Two roots are said to be like or unlike according as the indices of the roots are equal or unequal, whether the quantities under the radical sign are equal or not. Thus, 3 3 √ and 3√y are like roots; V and 3√y are unlike. 276. In this chapter will be considered the roots of numbers which are powers whose exponents are multiples of the indices of the roots. An even root of a number is one whose index is an even number; thus An odd root of a number is one whose index is an odd number; thus, 277. The Law of Signs of Roots of Quantities. From the law of signs in involution, 262, it is evident that: 1. Any even root of a positive number will have the double sign ±; because either a positive or a negative number raised to an even power is positive, 262. Thus, 1/16 +2, for (+2) = 16; 'Vaa, for (±a)* = a*. 2. Among the odd roots of a number there is at least one root of the same sign as the number itself. Thus, 2n+1,2n+1 -a5 = -a. In general, since (— a)2n+1— (− 1)2+1μ2+1——a2n+1; [ ? ? 263, 2; 262] 2n+1 √ (— α) 2n+1 = (− 1)(a) = — a. The principle stated in 2, when the radical is negative, may also be stated as follows: 3. An odd root of a negative number is minus the same root of a number which has the same absolute value. Thus, Hence, to find an odd root of a negative number, find the same root of the positive number which has the same absolute value, and prefix the negative sign to this root. 5. The even root of a negative number can not be taken; because no real number raised to an even power can produce a negative number. Such roots are called impossible. Thus, -3, since (+3)2 V-9 can not be +3 or 2n 9 and(-3)2 = 9. x, since (+x)3 = x2 and (— x)2 = x2. a, since (a)2n = a2n and (— a)2n = a2 Even roots of negative numbers can not be expressed in terms of numbers hitherto used, i. e., in terms of positive or negative integers, positive or negative fractions, or of positive or negative roots that can be found. The roots of numbers which are not powers with exponents which are multiples of the indices of the required roots and even roots of negative numbers will be discussed later. REMARK.-It has been shown above that a positive number which is the nth power of a number has at least one nth root and, when n is even, at least two; also that any negative number which is an odd power of a negative number has at least one odd root. It will be shown that any number has two square roots, three cube roots, four fourth roots, and five fifth roots; and in general it may be proved that any number has n, nth roots. 278. Principal Root.-1. The principal root of a positive number is its one positive root. Thus, 3 is the principal square root of 9, and 6 is the principal cube root of 216. 2. The principal odd root of a negative number is its one negative root. 3. It should be noticed at this point that the relation, holds for the principal nth root only. For, by the preceding article, the "a" has a values, the principal value being a. But, by the definition of a root, ("Va)" a for every ath root of a. Thus, √5o±5, if the negative root 5, as well as the principal root +5, is admitted; but = in case of the principal square root only. In the work which follows, the radical sign will be used to represent the principal root only, unless the contrary is expressly stated. Find the values of the principal roots indicated in the following examples: Using the definition of a root, express c as the root of the second member in each of the following equations: 17. x2-b. 18. 3-b2. 19. x13. 20. xb. 21. x2-bm. 22-26. Express as a root of the first member of each of the equations in 18–21. THEOREMS IN EVOLUTION In any case evolution is merely a special case of factoring, in which all the factors are equal. That is, the square root, the cube root, the fourth root, etc., are found by taking one of two, of three, of four, etc., equal factors, respectively of the given expression. Since even roots of negative numbers are not considered in this chapter and since the odd root of a negative number can be found by taking the like root of the same positive number (2277, 3), methods and rules for finding principal roots of positive numbers and expressions only will now be given. It is to be assumed in what follows that the radicand is a positive number or quantity and that the roots taken are principal roots. 279. THEOREM I.-The nth root of the product of several positive factors is equal to the product of the nth roots of each factor. Thus, ("Va • "vī • "ve)" = ("v@)" · ("vī)" · ("Ve)" := abc. [1263, 3] Hence equation (1) is true. 280. THEOREM II.-The nth root of the quotient of two quantities is equal to the quotient of the nth roots of the dividend and divisor. Thus, |