By the common measure we have the quotients 3, 7, 15, 1, ....; and by the rule for the formation of the terms we have 3 22 333 355 INVOLUTION AND EVOLUTION. .... THE term INVOLUTION has already been explained at pp. 118, 119, as the multiplication of several equal factors together; and it has been intimated that, for the case of the factors being binomial, a much more concise process will be given under the head of the Binomial Theorem. When there are more than two terms in the expression to be involved, it will, at least during elementary study, be better to have recourse to actual multiplication in the few cases that can arise, than to employ the Multinomial Theorem. EVOLUTION is the extraction of roots; that is, the inverse operation of raising powers, or of INVOLUTION. It constitutes but a very limited application of a general process, viz. that of the general solution of algebraic equations *. The rules usually given for Evolution are identical in substance with that here given for the square and cube roots; but in a form, though more briefly expressed, implying much more actual work in performing them. This rule is in fact only an expression of Horner's method of solving equations. CASE I. To extract any root of an expression composed of one single term. Divide the indices of all its factors by the index of the root to be extracted, and extract by any arithmetical process the root of the numerical coefficient. The continued product of all these roots is the root sought. !!! Thus the nth root of pambe is pabc"; and the cube root of 27a3b6ce is 27abc or 3a623 36 Of the signs of the results, it is only necessary to recollect that all odd roots of a negative quantity are real and negative; of all positive roots, real and positive; that all even roots of a positive quantity are real, and either positive or negative; whilst all even roots of a negative quantity are impossible or imaginary. This, however, is to be understood as applying to one individual root in each single case: since, besides these, there may be several other roots, wholly imaginary; and in all instances above the square root there actually are such roots. This view of the subject, however, belongs to a more advanced stage of the study of algebra. * An extension of the signification of the terms Involution and Evolution has been recently proposed by a distinguished mathematician, Professor De Morgan, viz, to the composition and resolution of algebraic equations. Involution and Evolution being particular cases of those general problems, the extension is perfectly justifiable; and in a treatise founded on this idea there would at least be the advantage of keeping those subjects together which were naturally connected with each other. It would, however, interfere too much with the existing arrangement of the work to adopt it here. To extract any root of a compound quantity. The breadth of our page will not allow us to exhibit an example of a higher root than the third; but we shall enunciate the rule for all cases, and give the form of work for the square and cube roots. 1. Write the given quantity, arranged according to the powers of some one letter * in the place of the dividend, and the curve to the right for the root, as in the arithmetical square and cube roots, pp. 66-72. 2. Make N columns to the left of the given expression, numbering them backwards from that expression as columns (1), (II), (III), .... (N). 3. Extract the Nth root of the first term of the given expression, and put that root in the column to the right of the curve. Denote, for the purpose of continuing the directions for working, this root by r, whatever that root may be. 4. Put 1 in column N; 1 xrin col. (N-1); (1 x r)r in column (N-2); and continue the process till the last result falls under the given expression: then evidently this result will be equal to the first term of the given expression. Subtract this, and bring down N terms for a dividend. 5. Form a new horizontal line as follows. Multiply 1 by r, place it in column (N-1), and add it to the previous result in that column; multiply this sum by r, and add it to the next column; multiply this sum by r, and add it to the next; and so on till the result falls in column (1). Then form a new horizontal line in the same manner, adding each product to the result above which it is written; but stop in each horizontal line one column sooner than in the preceding. We shall thus obtain a series of results, one in each column, preparatory to evolving the next term of the root. 6. With the expression in col. (1) as a divisor and the first term of the new dividend, find a new term of the quotient. This will be the second term of the root, which is to be used in forming the several columns, as before described. Proceeding thus, we shall obtain the successive terms of the root, if it be an exact root, or as many of them as we desire, if it be not exact. EXAMPLES. 1. Extract the square root a4 - 4a36 + 6a2b2 4ab3 + 64. Here the index of the root is 2, and the highest power of a is the fourth. Hence a2 = a2, and the condition is fulfilled. Also the powers follow regularly, and we can work with detached coefficients; but to show the identity of the methods, the work is put down here both ways. Also as n = 2, there will be two columns, and we have The rule cannot be applied except the first term is of a power exactly divisible by the index of the root; that is, for the nth root we must have the first term amn where m is an integer. which is worked in strict accordance with the rule, and it is evidently identical with that given for the square root of numbers, p. 67. The same example, by detached coefficients. Here a = a, and the condition of the applicability of the rule is fulfilled. Write it under the form a2 (1- : then we have to multiply the square root detached coefficients are 1 - 1; and the process will be as follows : in which the method of work is very simple and easy, and attaching the letters 226 }, or Ex. 3. Extract the cube root of a -6a5 + 21a2 - 44a3+63a2- 54a + 27. Here there are three columns, and the condition is fulfilled for the application of the rule. Whence 3-6+3 3-12-21-18+9, and attaching the powers, we have the root a2 2a + 3. The several terminated courses of operations are marked by dark lines immediately above their results. 4. The square root of a2 + 4a3b + 10a2b2 + 12ab3 + 964 is a2 + 2ab + 3b2. 5. To find the square root of a2 + 4a3 + 6a2 + 4a + 1, and of 1 + 4a + 6a2 + 4a3 + a2. 6. Extract the square root of a2 7. The square root of a2 - ab is a Ans. a2 + 2a + 1, and 1 + 2a + a2. 2a3 + 2a2-a + Ans. a2 -a+ b b2 2 8a 16a2 8. Find the square root of (a2 + x2) and of (x2 + a2). 9. The square root of a2 10. The cube root of a 11. The 4th root of 81a4 12. The 5th root of a5 2ab + 2ax +62 - &c. 2bx + x2, is a - b + x. 13a3+ 18a2 - 12a + 8, is a2 - a + 2. 216a2b2 - 96ab3 + 1664, is 3a 1004 + 40α3 - 80a2 + 80a 32, is a-2. 13. Find the square root of 1 - y2 and the cube root of 1 26. 14. Expand (a2- 2bx-x2) and a2 - æ2 into infinite series. 5 .... 5 .... In all these cases, however, the binomial theorem will be more convenient and effective. SURDS. [The student is advised to defer this subject till he has read some portion of the simple and quadratic equations.] SURDS are such quantities as have no exact root, and are usually expressed by fractional indices, or by means of the radical sign ✓. Thus, 3, or √3, denotes the square root of 3; and 23, or 3/22, or 3/4, the cube root of the square of 2; where the numerator shows the power to which the quantity is to be raised, and the denominator its root. The index may be put also in the form of a decimal, and is often so used. As for a writing a, or for a-, writing a- 6.333... See Definitions, p. 107. PROBLEM I. To reduce a rational quantity to the form of a surd. RAISE the given quantity to the power denoted by the index of the surd; then over or before this new quantity set the radical sign, and it will be of the form required. Note. When any radical quantity has a rational coefficient, this coefficient may be put under the irrational form, and the whole of the factors thereby brought under the symbol of radicality. Thus, instead of 3ab, we may put √√3ax3axb, or √9a2b; and so of others. EXAMPLES. 1. To reduce 4 and - 4 to the form of the square root. First, 42 = 4 × 4 = 16; then + 16 is the answer. 2. To reduce 3a2 to the form of the cube root. First, 3a2 × 3a2 × 3a2 = (3a2)3 = 27a; then we have 3/27a or (27a). 3. Reduce 6 to the form of the cube root. 4. Reduce - lab to the form of the square root. 5. Reduce + 2 to the form of the 4th root. Ans. (216) or √216. Ans. Ans. + (16). 7. Reduce a + x and x + a to the form of the square root. 8. Reduce ar to the form of the cube root. 9. Reduce (- 4a2 + b) / - ab to the form of the sixth root; and likewise to the form of the square root. 10. Transform (a - b) (a + b) √a2-b2 into its simplest form, and likewise represent them as radicals of the 3d, 4th, 5th, and 6th degrees. 11. Reduce (4-1) × 3 to the form of a cube root; and then reduce + bc +1-bc + 1 to the simplest form it admits of. PROBLEM II. To reduce quantities to a common index. 1. REDUCE the indices of the given quantities to a common denominator, and involve each of them to the power denoted by its numerator; then 1 set over the common denominator will form the common index. Or, 2. If the common index be given, divide the indices of the quantities by the given index, and the quotients will be the new indices for those quantities.Then over the said quantities, with their new indices, set the given index, and they will make the equivalent quantities sought. EXAMPLES. 1. Reduce 3 and 53 to a common index. Here, and = and 2 Therefore 30 and 510 = (35) and (52) 10 = 10/35 and 10/52 = 10/243 and 10/25. |