SURDS .] ALGEBRA fraction having 3 for a denominator, which will be the fraction %; therefore a 2 = s = ^/a 6 . 38. II. Reduction of Surds of different denominations to others of the same value and of the same denomination. Rule. Reduce the fractional exponents to others of the same value, and having the same common denominator. 3 _ 1 2 Ex. Reduce ,Ja and Jb 2 , or a 2 " and V s to other equi valent surds of the same denomination. The exponents , f, when reduced to a common deno minator, are f and; therefore the surds required are 46 _ 6 _ . o^ and b G , or Jo? and Jb* . 39. III. Reduction of Surds to their most simple terms. Rule. Reduce the surd into two factors, so that one of them may be a complete power, having its exponent divi sible by the index of the surd. Extract the root of that power, and place it before the remaining quantities, with the proper radical sign between them. Ex. 1. Reduce ^48 to its most simple terms. The number 48 may be resolved into the two factors 16 and 3, of which the first is a complete square; therefore Ex. 2. Reduce v / 98a%, and its most simple terms. . i _ o i First, tjQ8a*x = (7 2 a 4 x 2x) ir = < a- x (2a?) T = 7a 2 3 1 3 40. IV. Addition and Subtraction of Surds. Rule. If the surds are of different denominations, reduce them to others of the same denomination, by prob. 2, and then reduce them to their simplest terms by last pro blem. Then, if the surd part be the same in them all, annex it to the sum or difference of the rational parts, with the sign of multiplication, and it will give the sum or difference required. But if the surd part be not the same in all the quantities, they can only be added or subtracted by placing the signs + or - between them. Ex. 1. Required the sum of J27 and */&. By prob. 3 we find ^27 = 3^/3 and ^/48 = 4 ^3, there fore J27 + ^48 - 3 J3 + 4 J3 - 7 ^3 . 3 ._ 3 __ Ex. 2. Reuired the sum of 3 * and 5 -fa . 3 J/i3 t/t-f J/2 and 5 therefore 3 J/f+ 5 ^ = = 5 = V Ex. 3. Required the difference between v /80a 4 ^ and = (4-V x 4a 2 ^/5^, and (2 2 aV x 5x)~*= 2ax J5x; therefore v 3 (4a 2 2ax] Jox . 41. V. Multiplication and Division of Surds. Rule. If they are surds of the same rational quantity, add or subtract their exponents. But if they are surds of different rational quantities, let them be brought to others of the same denomination, by prob. 2. Then, by multiplying or dividing these rational quantities, their product or quotient may be set under the common radical sign. Note. If the surds have any rational coefficients, their pro duct or quotient must be prefixed. 3 _ 5 Ex. 1. Required the product of ^/a 2 and ^ 3_ 6 ._ 2 ri 0.,.* IS 1
ft 3? _ a l 5
, Ans. .E x. 2. Divide ^/a 2 i 2 by Ja + b . These surds, when reduced to the same denomination, xre (a 2 b 2 } 5 and (a + b}^. Hence ~y a ~ = l^ a ~ - | V+ & (a+6)V 42. VI. Involution and Evolution of Surds. The powers and roots of surds are found in the same manner as any other quantities, namely, by multiplying or dividing their exponents by the index of the power or root 3 2. required. Thus, the square of 3 J3 is 3 x 3 x (3) 3 = a - t - 9 1/9. The nth power of x m is x m . The cube root of 1-111- J_L - N /2 is - (2) = ^ *iJ2> anc ^ the n ^ root ^ xm xmn 43. The reduction of quadratic surds is facilitated by the following considerations, which appear hardly to require demonstration : v- 1. ^/a cannot = 6 + Jc, when ,Jc is a surd. 2. a+ ,Jb cannot = c+ Jd when ,Jb, Jd arc unequal surds. 3. a cannot = *Jb *Jc when ,Jb, ,Jc are surds not involv ing the same irrational part, ^2 and ^3 for example. 4. ,Ja cannot equal ,Jb + ,Jc when all are surds not in volving the same irrational part. Note. The irrational part of ^/S, for instance, is ^2, for ^8 = 2^/2. 44. For example, we extract the square root of a binomial surd such as 28 + 10 ^3 in the following way : Let ^/28 + 10 ^3 = x + y, where one or both of x and y must be a surd. Then 28 + 10 J3 = a? + 28 = 2 + ^ 2 , 10^/3 = 2^, or No. 2 above would be violated. Hence ^28-10^3= Ja? + y*-2xy And A y784-300-^ 2 -7/ 2 , or a 2 + / = 28
- = 0, y= *Jo
and 5 + ^/3 is the root required. Additional Examples in Surds. Ex. 1. Add together ^ , _L_ , -^-L- , and __L_ , I 12 1 + V2 + 1-V2 == 1^2 = L. V3 + 1 V3- ^/3 - 2 is the sum required. Ex. 2. Find the difference between a + x+*/(a+x) ___ -1 3-1" ^ The former is The latter is the square root of V(a+a;) // , 7? i- c -> f t V(+a-)-l a+jc
. . the difference reauired is 0.
