The following tables exhibit other applications of the rules: The powers of such an expression as a 1 are therefore alternately real and imaginary, and are positive and negative in pairs. (a + b√ − 1) (c + d√ − 1) Let the roots of the equation ax2+ bx + c = 0 be impossible, that is, let 4ac be negative and equal to k2. Its roots, as derived from the rules estaplished when b2 - 4ac was positive, are k2 or -1. √ – 1. k2 4a b2 k2 4a 4a ordinary rules be applied, produce the same results as the roots. They are thence called imaginary roots, and we say that every equation of the second degree has two roots, either both real or both imaginary. It is generally true, that wherever an imaginary expression occurs, the same results will follow from the application of these expressions in any process as would have followed had the proposed problem been possible and its solution real. When an equation arises in which imaginary and real expressions occur together, such as a + b −1 = c + d1, when all the terms are transferred on one side, the part which is real and that which is imaginary must each of them be equal to nothing. The equation just given when its left side is transposed become a-c+b-d-i = 0. Now, if b is not equal to d, let b-de; then a-c+e- 1 = 0, +c, in which, if 4ac – b be substi- and Za -a and pre The imaginary expression b have this the negative expression resemblance, that either of them occurring as the solution of a problem indicates some inconsistency or absur dity. As far as real meaning is concerned, both are equally imaginary, since 0 a is as inconceivable as√-a. What, then, is the difference of signification? The following problems will elucidate this. A father is fifty-six, and his son twenty-nine years old: when will the father be twice as old as the son? Let this happen x years from the sent time; then the age of the father will be 56x, and that of the son 29; and therefore, 56 + x = 2 (29 + x) = 582x, or x = This result is absurd; nevertheless, if in the equation we change the sign of x throughout it becomes 56-x-58-2x, This equation is the one belonging to the problem: a father is 56 and his son 29 years old; when was the father twice as old as the son? the answer to which is, two years ago. In this case the negative sign arises from too great a limitation in the terms of the problem, which should have demanded how many years have elapsed or will elapse before the father is twice or x=2. as old as his son? 2. and the roots of the second are imagi There nary, if those of the first are so. is, then, this distinct difference between the negative and the imaginary result. When the answer to a problem is negative, by changing the sign of x in the equation which produced that result, we either discover an error in the may method of forming that equation or show that the question of the problem is too limited, and may be extended so When the answer to a problem is imaas to admit of a satisfactory answer. ginary this is not the case. CHAPTER XI. On Roots in general, and Logarithms. THE meaning of the terms square root, cube root, fourth root, &c. has already been defined. We now proceed to the difficulties attending the connexion of the roots of a with the powers of a. The following table will refresh the memory of the student with respect to the meaning of the terms : Name of x. Square Root of a Cube Root.... Fourth Power Fourth Root... Fifth Root. The different powers and roots of a have hitherto been expressed in the fol lowing way: Powers a2 a a1 a5 am am+n, &c. Roots Va Va Va which series are connected together by the following equation, (ā)" = a. There has hitherto been no connexion between the manner of expressing powers and roots, and we have found no properties which are common both to powers and roots. Nevertheless, by the extension of rules, we shall be led to a method of denoting the raising of powers, the extraction of roots, and combinations of the two, to which algebra has been most peculiarly indebted, and the importance of which will justify the length at which it will be treated here. Suppose it required to find the cube of 2a2 b3; that is, to find 2a2 b3 x 2a2 b3 x 2a2 b3. The common rules of multiplication give, as the result, 8a6 b9, which is expressed in the following equation, = 8a6 b9 (2a2 b3)3 and the general rule by which any single term may be raised to the power whose index is n, is, raise the coefficient to the power n, and multiply the index of every letter by n, that is, (ar bi cr)n = anp bng cnr. In extracting the root of any simple term, we are guided by the manner in which the corresponding power is found. The rule is, extract the required root of the coefficient, and divide the index of each letter by the index of the root. Where these divisions do not give whole 'numbers as the quotients, the expression whose root is to be extracted does not AB = /A./B. For, raise 3⁄4Ã3/B to the third power, the result of which is A/B/A/B × √ √ B, or ANANA B /B/B, or AB. In the same way it may be proved generally, that ABC=A√√√√C. The most simple way of representing any root of any expression is the dividing it into two factors, one of which is the highest which it admits of whose root can be extracted by the rule just given. For example, in finding√16ab7c we must observe that 16 is 8×2, a1 is a3× a, 67 is b× b, and the expression is 8a3 b6 x 2abc, the cube root of which, found by extracting the cube root of each factor, is 2ab22abc. The second factor has no cube root which can be expressed by means of the symbols hitherto used, but when the numbers which a, b, and c stand for are known, 2abc may be found either exactly, or, when that is not possible, by approximation. We find that a power of a power is found by affixing, as an index, the product of the indices of the two powers. Thus Ja Ja Ja root of a is of which is α= Ja Ja Ja, the square a Ja Ja, the cube root a. This is the same as admit of the extraction without the introduction of some new symbol. For example, extract the fourth root of 16a12 b8 c2, or find √16a12b8 c. Thea, and generally expression here given is the same as the following: 2as b2 cx 2a3 b2 cx 2a3 b2 cx 2a3 b2 c, or (2a3 62 c), the fourth root of which is 2a3 bc, conformably to the rule. Any root of a product, such as AB, may be extracted by extracting the root of each of its factors. Thus, Again, when a power is raised and a done first. Thus 3/2 is the same thing root extracted, it is indifferent which is as (3). "For since ̈ a2= a × a, the cube root of each of these factors, that is cube root may be found by taking the 2 a = Vā× Vā= (Vā)3, and generally mp mp mp or, to divide one power of a quantity by Apply the last two rules, and it appears another, subtract the index of the divisor from that of the dividend, and make the difference the index of the result. Suppose it required to find It is evident that a qm m m m m n Ха = α n n 3 = that (a) = 3, and √=¿ na And the rule is;-to raise one power of a quantity to another power, multiply the indices of the two powers together, and make the product the index of the result. = a. Similarly (a) All these rules are exactly those which a", and so on. Therefore Again to find a = α : m this be . Then &o. a". m a = or ay a == y n' y Again to find m m " and nq' (a) == or am-n. The only cases which have been considered in forming this rule are those in which m is greater than n, being the Therefore only ones in which the subtracted indicated is possible. If we apply the rule to any other case, a new symbol is proα = a duced, which we proceed to consider. For example, suppose it required to find = ng ("). If we apply the rule, we find the |