The greater a is with respect to b, the more nearly is the true value obtained by division; [for the first remainder divided [155. In extracting the square or cube root of a vulgar fraction the rule stated in Art. 145 may be followed; but it is generally preferable to convert the vulgar fraction into a decimal, and then extract the root. Thus let the cube root of 5, or 11 2 be required. Now, if the rule of Art. 145 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.] 156. 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. (See Note 6. Appendix 1.) SURDS. 157. 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. Thus a = 3 √ √a3, &c. and a + x = In the same manner, the form of thus (a + x) = (a + x) = (a + x)* &c. m (a + x)m. any surd may be altered; 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. 158. The coefficient of a surd may be introduced under the radical sign by first reducing it to the form of the surd, and then multiplying according to Art. 144. Exs. a√x=√a2 × √ x = √ œ2 x ; ay* = (a2y3)3 ; x√2a − x = √ 2 a x2 − x3 ; a × (a− a) 4 = {a2 × (a − x)3} 1 ; 4√/2=√16x2 = √32. 159. 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. Thus √ a2 - ax = a3√ a − x ; √ a3 − a3 x = a √ a − x ; √/60 = √/4x 15 = 2√/15; a2 160. 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. Thus a√x±b√x=(a±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. 159. Thus, Let the sum of √3ab and √3b be required. √3a2b+ √3b=a√/3b + √3b = (a + 1) √3b. 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. 161. 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 1 n Thus √ax√bax b1 = (ab)" (Art. 144) = √ab. = an √2 × √3 = √6; (a + b)3 × (a - b)3 = (a2 — b2)3 ; 162. If the surds have coefficients, the product of these coefficients must be prefixed. Thus a√xxb√y=ab√xy. 163. 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. also (a+x)3× (a− x)} = {(a + x) (a − x)3 } 3. 164. 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. 163. Ex. (a2 — x2)1 × (a − x )3 = (a2 — x2)} × (a− x); = again 2 × 3 = 28 × 3a = (8 × 9)3 = (72)'. 165. 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. (See Art. 144, and Note 1. Appendix 1.) Ex. √2x2= 21+1 = 21. DIVISION.. 166. 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. 167. If the indices have not a common denominator, reduce them to others of the same value with a common de-. nominator, and proceed as before. 2 Ex. (a2 —¿2) ÷ (a3 − x3)} = (a3 − x2)3 ÷ (a3 −x3)3 168. If two surds have the same rational quantity under the radical signs, their quotient is obtained by making the difference of the indices the index of that quantity. Thus an divided by am, or a divided by amn m amn m-n (Art. 157), that is is equal to ama; because these quantities, raised n , INVOLUTION AND EVOLUTION. [169. Any power of a surd is found by multiplying the index of the surd by the number which expresses the power. 170. Any root of a surd is found by dividing the index of the surd by the number which expresses the root. 1 Thus √(√α) = √√a^ = am m n = amn; because each of these quantities raised to the mth power will produce an. It will be seen that the rules hitherto required for the management of surds are simply those which apply to quantities raised to powers expressed by Fractional Indices. See Note 1. Appendix. 1.] TRANSFORMATION OF SURDS. [171. Having given a quantity containing quadratic surds, to find another quantity which, multiplied into the former, shall produce a rational result. 1. If the given quantity be a simple surd, as 3√a, the multiplier required is √a, which gives the product 3a, a rational quantity. 2. If the given quantity be a binomial surd, as √a+√b, then the multiplier required is √a -√b, and the product is a - b. √a+√b 3. If the quantity be a trinomial, as √a + √b+ √c, first multiply by √a+√b-√c, which gives (√α + √√õ)2 - (√c)2, or a+b-c+2√ab. Next multiply by a+b-c - 2√ab, and the product is (a + b − c)2 - 4ab. Therefore the multiplier required is (√a+√b-ve) × (a + b − c −2√/ab). The use of this proposition is to enable us without much labour to find the values of fractions which have irrational denominators. Thus, suppose the actual value of were required to 7 places of decimals; if we 1 √3+√2 |