11 Thus let the cube root of 54, or be required. Now, if the rule of Art. 151 be applied to this case, the cube root of 11, and the cube root of 2, must be found to a certain number of places of decimals, and then the long division of the one root by the other must be effected: whereas, if 5 be, first of all, converted into a decimal, viz. 5.5, one single extraction of the cube root completes the whole process. Another method is, to multiply the numerator and denominator by such a quantity as will make the latter a perfect cube, and then apply the rule of Art. 151. 162. In extracting either the square or cube root of any number, when a certain number of figures in the root have been obtained by the common rule, that number may be nearly doubled by division only. I. The square root of any number may be found by using the common Rule for extracting the square root until one more than half the number of digits in the root is obtained; then the rest of the digits in the root may be determined by Division. For, let N represent the number whose square root, consisting of 2n+1 digits, is required; a ......... .... the first n + 1 digits of the root found by the common Rule, with n cyphers annexed; that is, N – a3, (which is the remainder after n + 1 digits of the root are found) divided by 2a will give the rest of the root required, increased by x2 2 a Now, since x contains n digits, x2 has 2n at most (Art. 155). But, by the supposition, 2a has 2n + 1 digits; therefore that is, if the quotient of (N-a2) ÷ 2a be taken for x, the error is less than 1. Hence it appears, that if n + 1 digits of a square root are obtained by the common Rule, n digits more may be correctly obtained by Division only. Ex. Required the square root of 2 to 6 places of decimals. 2.0000...(1.414 1 24) 100 96 281) 400 281 2824) 11900 2828) 604000 (213 5656 3840 2828 10120 8484 1636 .. the root required is 1.414213...... II. When a cube root consists of 2n+2 digits, and n+2, one more than half the number, have been found by the ordinary rule, the rest can be found by dividing by the trial divisor. Let a + b be the cube root, where a consists of n + 2 digits, and n cyphers, n digits. a3 + 3a2b+3ab2 + b3 the quantity whose root is required. Then after a has been found, we have remainder = 3a2b+3ab2 + b3 ; that is, the last n figures of the root may be correctly obtained by division only. 163. REDUCTION OF SURDS. A rational quantity may be reduced to the form of a given surd by raising it to the power whose root the surd expresses and affixing the radical sign. 3 Thus a = = 2 m m m a3, &c. and a + x = (a + x) = (a + x)TM. In the same manner, the form of any surd may be altered; thus (a + x)3 = (a + x)2 = (a + x) &c. The quantities are here raised to certain powers, and the roots of those powers are again taken; therefore the values of the quantities are not altered. 164. The coefficient of a surd may be introduced under the radical sign by first reducing it to the form of the surd, by the last Art., and then multiplying according to Art. 149. Exs. a√x=√ã3× √x = √ a2x; 165. ay* = (a2y3); x√2 a 00= √2αx2 - x3; Conversely, any quantity may be made the coefficient of a surd, if every part under the sign be divided by this quantity raised to the power whose root the sign expresses. b2 166. ADDITION AND SUBTRACTION OF SURDS. When surds have the same irrational part, their sum or difference is found by affixing to that irrational part the sum or difference of their coefficients. √x; Thus a√x±b√x= 10√/3±5√/3=15/3, or 5 √3. If the proposed surds have not the same irrational part, they may sometimes be reduced to others which have, by Art. 165. Thus, Ex. 1. Let the sum of √3a2b and 36 be required. Since √3a2b√aa × √3b=a √3b, = ..√3a2b+ √3b =a√3b+ √3b= (a + 1) √3b. b/8ab=b8a. *b=2abb, √125a*b* = − √125ab3, √b=-5a2b√b; .. the sum required = ab /b. If the proposed surds cannot be reduced to others which have the same irrational part, then they must be connected together merely by the signs + and MULTIPLICATION OF SURDS. 167. If two surds have the same index, their product is found by taking the product of the quantities under the signs and retaining the common index. 1 Thus √ a × √b = a3× b" = (ab)" (Art. 149) = √ab. √2×√3 =√6; (a + b)3 × (a − b)3 = (a2 — b2)}; If the surds have coefficients, the product of these coefficients must be prefixed. Thus a√xx b√y=ab√xy. 168. If the indices of two surds have a common denominator, let the quantities be raised to the powers expressed by their respective numerators, and their product may be found as before. Ex. 9 x 3* = (23)! × 3h = 81 x 34 = (24); also (a + x)3× (a − x)a = {(a + x) (a − x)3}3. 169. If the indices have not a common denominator, they may be transformed to others of the same value with a common denominator, and their product found as in Art. 168. Ex. (a2 – x2)1× (a − x)3 = (a2 − x2)‡ × (a − x)i = 3 again 2a × 3 = 20 × 3 = (8 × 9)* = (72)* . 170. If two surds have the same rational quantity under the radical signs, their product is found by making the sum of the indices the index of that quantity. m 1 1 m n m+n Thus Vaxaan = a a = añ × am = ɑmn × ɑmn = ɑ mn; (see Arts. 149, and 150.) Ex. √2x2 = 2a × 2} = 21+1 = 21. DIVISION OF SURDS. 171. If the indices of two quantities have a common denominator, the quotient of one divided by the other is obtained by raising them respectively to the powers expressed by the numerators of their indices, and extracting that root of the quotient which is expressed by the common denominator. 172. If the indices have not a common denominator, reduce them to others of the same value with a common denominator, and proceed as before. Ex. (a2 − x2)3 ÷ (a3 − ∞3)* = (a − x2)3 ÷ (a3 − x3)3 |