EVOLUTION.] ALGEBRA 529 ing in what manner the terms a, b, and c are deduced from this expression for the cube of their sum, we also see the reason for the common rule for extracting the cube root in numbers. Let it be required to find the cube root of 13312053, where the root will evidently consist of three figures ; let us suppose it to be represented by a + b + c, and the operation for finding the numerical values of these quantities may stand as follows : 13312053(200 = a = 8000000 30 = 6 120000 5312053 3a5= 18000 tf= 900 3a 2 + 3ab + tf = 138900 41 67000 = (3a 2 + 3a& + 6 2 ) 2 = 158700 )c = 4830 c 2 = 49 237 is the root required. 1145053 2 = 163579 1145053 = 3(a ]c III. To extract any other root. Eange the quantity of which the root is to be found, according to the powers of one of its letters, and extract the root of the first term; that will be the first member of the root required. Involve the first member of the root to a power less by unity than the number that denominates the root re quired, and multiply the power that arises by the num ber itself ; divide the second term of the given quantity by the product, and the quotient shall give the second member of the root required. Find the remaining members of the root in the same manner by considering those already found as making one term. 24. In the preceding examples, the quantities whose roots were to be found have been all such as could have their roots expressed by a finite number of terms; but it will frequently happen that the root cannot be otherwise assigned than by a series consisting of an infinite number of terms. The preceding rules, however, will serve to de termine any number of terms of the series. Thus, the /v2 /v square root of a 2 + x 2 will be found to be a + - ? ; + 2a 8a 3 16a 5 ~ thus, 5x 8 128a 7 x 3 + &c., and the cube root of aP + x 3 will stand 10x 12 243a u + &c. But as the ex- traction of roots in the form of series can be more easily performed by the aid of the binomial theorem, we shall refer the reader to the section where this subject is resumed. Additional Examples. 3 Ex. 1. Write down the square root of a; 4 - 2# 3 + - x 2 - 2 -x + , which is given as a perfect square. J 10 Since the square contains 5 terms, the root must con tain 3. Of these the first is x 2 on account of # 4 , the second - x on account of 2# 3 and the third - on account of . 4 16 But as the last tenn but one of the square is - , and the last term but one of the root also - , the last term of the root must be + . 1 > .. x- - x + - is the root required. Ex. 2. Extract the square root of 25# 4 + IGy 4 - 6xy (5# 2 + 4y 2 ) + 49.CV 2 . We must first arrange the square in terms of some one quantity (say x). The first term of the square is 25x A , which gives 5x 2 as the first term of the root. The second term of the square, - 3Qx 5 y gives - 3xy as the second term of the root. The last term IGy* gives 4y 2 ; which, since the last term but one is - , leads to the root 5 2 - 3xy + 4y 2 . Ex. 3. Extract the cube root of 8x* - 36x 5 + 663* - 63# 3 + 33a 2 - 9* + 1 . Since there are seven terms in the cube, there must bo three terms in the root. The first is 2# 2 , the second - 3x, the third 1, as will be seen at once by examining the cube of p-q+l, viz., p 3 - 3p 2 q + ... -3q+l . These examples have been solved by the assumption that the root is capable of extraction without leaving a remainder. When this is not the case, or when there is no certainty that it is so, the only resource is to work the example through, abbreviating the process by the aid of detached coefficients. Ex. 4. Extract the square root of 4:X Q + l2x 5 y + 5x*y- 2x*y 3 + 7a; i y - 2xy 5 + y. The work is written thus : 4 + 12 + 5-2 + 7-2 + 1(2* 3 + 3x^ -xy^ + y* 4 + 3) 12 + 5 12 + 9 4 + 6-1) -4-2 + 7 -4-6+1 4 + 6-2 + 1 ) 4 + 6-2 + 1 Ex. 5. Extract the cube root of 27^ 27 x^y 45.# 4 3/ 2 + 35x 3 y 3 + We have 27-27-45 + 35 + 30-12-8(3* 2 -*y 27 27 ) -27-45 + 35 -27+ 9-1 27 - 18 + 3) - 54 + 36 + 30 - 12 - 8 I--54 + 36- 6 -] + 36-12 -8J SECT. III. FEACTIONS. 25. In the operation of division, the divisor may be some times greater than the dividend, or may not be contained in it an exact number of times : in either case the quotient is expressed by means of a fraction. There can be no difficulty, however, in estimating the magnitude of such a quotient ; if, for example, it were the fraction , we may consider it as denoting either that some unit is divided into 7 equal parts, and that 5 of these are taken, or that
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