528 ALGEBRA [INVOLUTION AND If we had begun by tlircnving the expression into the form of (by + ~Pz + Tix)- -f- &c., a resulting condition would have been 6c>P 2 . The four conditions arc con sequently a?;>R 2 , ac>Q 2 , &c>P 2 , ale + 2PQR >P 2 + iQ 2 + c-R 2 . Results of this kind arc higher analvsis. )f the utmost value in the Evolution. 23. Evolution is the reverse of involution, or it is the method of finding the root of any quantity, whether simple or compound, which is considered as a power of that root : hence it follows that its operations, generally speaking, must be the reverse of those of iuvohition. To denote that the root of any quantity is to be taken, the sign J (called the radical sign] is placed before it, and a small number placed over the sign to express the denomination of the root. Thus f/a denotes the square root of a, 4/ a ^s cube root, f/a its fourth root, and in general, j/a its nth root. The number placed over the radical sign is called the index or expojicnt of the root, and is usually omitted in expressing the square root: thus, either f/a or ,Ja denotes the square root of a. Case 1. When roots of simple quantities are to be found. Rule. Divide the exponents of the letters by the index of the root required, and prefix the root of the numeral coefficient; the result will be the root required. Noie 1. The root of any positive quantity may be either positive or negative, if the index of the root be an even number; but if it be an odd number, the root can be positive only. 2. The root of a negative quantity is also negative when the index of the root is an odd number. 3. But if the quantity be negative, and the index of the root even, then no root can be assigned. Ex. Required the cube root of 125a 6 x 9 . Here the index of the root is 3, and the root of the co efficient 5, therefore f/125a 6 x g = 5a 2 x 3 , the root required; and in like manner the cube root of- 125a 6 .r 9 is found to be - 5a^x 3 . The root of a fraction is found by extracting the root of both numerator and denominator. Thus the square , 4a 2 x 4 . Zay? root of -TT-T is r- Qb z y a 3by 3 Case 2. When the quantity of which the root is to be extracted is compound. I. To extract the square root. Range the terms of the quantity according to the powers of one of the letters, as in division. Find the square root of the first term for the first part of the root sought, subtract its square from the given quantity, and divide the remainder by double the part already found, and the quotient is the second term of the root. Add the second part to double the first, and multiply their sum by the second part ; subtract tho product from the remainder, and if nothing remain, the square root is obtained. But if there is a remainder, it must be divided by the double of the parts already found, and the quotient will give the third term of the root, and so on. 3 y 1 Ex. Required the square root of # 4 -2.r ! + x- 1
- lU
x - xj - 2X 3 + x~ x*- 2x + -)--- + f- 4/2 210 x- x 1 1 X 4 To understand the reason of the rule for findisg tho square root of a compound quantity, it is only necessary to involve any quantity, as a + b + c, to the second power, and observe the composition of its square; for we have (2a + b)b and 2ac + 2bc + c 2 = (2a + 2b + c)c, therefore, (a + b + c) 2 = a 2 + (2a + b)b + (la + 2b + c)c; and from this expression the manner of deriving the rule is obvious. As an illustration of the common rule for extracting the square root of any proposed number, we shall suppose that the root of 59049 is required. Accordingly we have (a + I + c) 2 = 59049, and from hence we are to find the values of a, l>, and c. 59049(200 = a) a 2 = 200x200 = 40000 40 = 6 I Hence 243 is the root o _ j required. 2a = 40o l9049 6= 40 2 + 6 = 440 2a + 26 = 480 1449 c= 3 II. To extract the cube root. Range the terms of the quantity according to the powers of some one of the letters. Find the root of the first term, for the first part of the root sought ; subtract its cube from the whole quantity, and divide the remainder by three times the square of the part already found, and the quotient is the second part of the root. Add together three times the square of the part of the root already found, three times the product of that part and the second part of the root, and the square of the second part; multiply the sum by the second part, and sub tract the product from the first remainder, and if nothing remain, the root is obtained ; but if there is a remainder, it must be divided by three times the square of the sum of the parts already found, and the qiiotient is a third term of the root, and so on, till the whole root fs obtained. Ex. Required the cube root of a 5 + 3a-x + 3ax- + x 3 . a 3 + 3a-x + Sax" 2 + x*(a + x, the root required. 3a 2 + Sax + x- 3aa,- 2 The reason of the preceding rule is evident from the composition of a cube ; for if any quantity, as a + b + c, be raised to the third power, we have (a + 6 + c) 3 = a 3 + (3a 2
+ Sab + b z )b + {3(a + 6) 2 + 3(a + b)e + c-}c, and by consider-
