EXTRACTION OF ROOTS. See EVOLUTION. The roots which have in practice to be most frequently extracted are the square and cube roots. It is proposed to explain the rule for their extraction as it is given in books of arithmetic. And first of the square root. The square of a + b is 2ab + and we may obtain the rule by observing how a b may be deduced from it. Arranging the expression according to powers of some letter a, we observe that the square root of the first term is a.
Subtract its square from the expression, and the remainder is 2ab Divide 2ab by 2a, and the result is b, the other term in the root. .Multiply 2a b by b, and sub tract the product from the remainder. If the operation does not terminate, it shows that there is another term in the root. In this case, we may consider the two terms a b already found as one, and as corresponding to the term a in the preceding opera tion; and the square of this quantity having been by the preceding process subtracted from the given expression, we may divide the remainder by 2(a + b) for the next term the root, and for a new subtrahend multiply 2(a + b) + the new term, by the new term; and the process may be repeated till there is no remainder. The rule for extracting the square root of a number is an adaptation of this algebraical rule. In fact, if the number be expressed in terms of the radix of its scale, it is seen to be a concealed algebraical expression of the order we have been considering. Thus, N = . . . q. The number 576 in the denary scale may be written 5 X + 7 X 10 + 6; and treat ing it as an algebraical expression, we should find its root to be 2 X 10 4, or 24_ The only part of the arithmetical rule now requiring explanation is the rule of pointing. As every number of one figure is less than 10, its square must be less than 10'; gener ally, every number of n figures is less than 10n (which is 1 followed by n ciphers); but also every number of n figures is not less than and therefore its square is not less than is the smallest number of 2n — 1 figures. Also, is the
smallest number of 2u 1 figures. It follows that the square of a number of n figures has either 2n or 2n — 1 figures. If, then, we put a point over the units place of a num ber of which the root is to be extracted, and point every second figure from right left, the number of points will always equal that of the figures in the root. If the num ber of figures be even, the number will be divided into groups of two each; if odd, the last group will contain only a single figure.
The rule for the extraction of the cube root of a number is deduced from that for the extraction of the cube root of an algebraical expression in the same way as in the case of the square root. The cube of (a b) is b Hence the rule in algebra. Arrange the expression according to descending powers of a, the cube root of the first term is a, the first term of the root. Subtract its cube from the expression, and bring down the remainder. Divide the first term by and the quotient is .5, the second term of the root. Subtract the quantity If there is no remainder, the root is extracted. If there is, proceed as before, regarding b as one term, corresponding to a in the first operation. Let, for example, a + b = a', then 30 9 is the new trial divisor. If c be the new term or third figure of the root, then the quantity to be subtracted to get the next remainder is 3cil + e, and so on till there is no remainder. The rule of pointing in the extraction of the cube root may be proved, as in the case of the square root, by showing that the cube of a number of /I figures contains 3n, 3n — 1, or 3n — 2 figures; and, therefore, if we put a point over the units place, and on each third figure, we shall have as many periods as there are figures in the root.
It may be observed that a rule for the extraction of any root of a number may be got from considering how, from the expansion of a b to the nth power, or a^ + +, etc., the root a + b is to be obtained. See EVOLUTION and INVOLUTION.