4. Reduce (s) and (%) * to equivalent surds having a common index. Ans. (**)'s and (i )*'s. (a) and ( )* to equivalent surds hav (*)* and (tot)*: 5. Reduce ing a common index. Ans. 88. To add or subtract surds. RULE. Reduce the surds to their simplest form; then if they have the same radical quantity in each, the sum of the coefficients prefixed to this radical will be their sum; and the difference of the coefficients prefixed to the radical will be their difference. But if they have different radi. cal quantities, their sum can only be indicated by placing the sign plus between them, and their difference by placing the sign minus between them. The reason of this rule is obvious, for the radical quantity may be represented by a letter, and then the rule will be identical with that of addition and subtraction in algebra. EXAMPLE. What is the sum and difference of 288 and 128. Here 288=7144x2=12V2, and w 128 =V64 x2=812; hence their sum is 2012, and their difference is 4N2. 1. Find the sum and difference of 3V 32 and 2w 162. Ans. sum 3072, diff. 6.12. 2. Find the suín and difference of 3/54 and 3/250. Ans. sum 143/2, diff. 43/2. 3. Find the sum and difference of 324a4 and/192a. Ans. sum (2a+4)33a, diff. (20—4) 3/3a. 4. Find the sum and difference of N 80 and W 45. Ans. sum 7V5, diff. v5. 5. Find the sum and difference of (36a3); and (98a). Ans. sum (6a1a+7/2a), diff. (6avā-712a.) 6. Find the sum and difference of (1000a")and (300a3). Ans. sum (10a2v 10a+10aw 3a), and diff. 10a2VI0a -10aw3a. TO MULTIPLY SURDS. 89. RULE. Reduce the surds, if necessary, to a common index, then multiply the coefficients together for a coefficient, and the surd quantities together for the surd, over which place the common index. EXAMPLE. Multiply 3/10 by 2 12. 3x2=6, the coefficient, and 3/10x3/12=3/120. :: 63120 is the result; which, however, can be simplified; for 120=3/8x15=23/15; hence the quantity in its simplest form is 12/15. EXAMPLE 2. Multiply vã by 26. Ans. 30V 10. 2. Multiply (18)3 by 577. Ans. 1079. 3. Multiply V10 by 3/15. Ans. (233255); or 9225000. 4. Multiply 5a3 Ans. a?!. 5. Multiply (a+b)# by (a-b} Ans. (a*_62) 6. Multiply a" by a. by dat m mstur Ans. a ns TO DIVIDE SURDS. 90. Rule. Reduce the quantities, if necessary, to a common index, then divide the coefficients and the surd quantities separately as in rational quantities. EXAMPLE. Divide ab ac by 6/bc. Here ab=b=a, the coefficient, and ac+bc=, the surd :. the quotient is app 7. EXAMPLE 2. Divide 3V ac by 23/dc. Here the quantities reduced to a common index become 3(023)#, and 2 (6?co) the coefficient of the quotient 3 26 is , and the surd (a)t=C) which reduced to its simplest form is ž (230°c), and hence the quotient is bc Otherwise, 3. Jac zaled zake_3(23cg)_3 1a9c3ja 2V027c2(020)7–2 620 33 1. Divide 10 V 27 by 2V3. Ans. 15. 2. Divide by 3/6. Ans. 3. Divide 43/18 by 2/9. Ans. 232. 4. Divide vã by 3/ab. Ans. Vabar . 5. Divide Va3 by Na%. Ans. gaš. 6. Divide (a?—62) by (a−b)]. Ans. (a+b)?! (236c). INVOLUTION OF SURDS. 91. RULE. Raise the coefficient of the surd to the required power, and then multiply the exponent of the surd by the exponent of the power. EXAMPLE. Find the third power of 2vac. Here we raise 2 to the third power, which gives 8, and then multiply }, the exponent of the surd, by 3, the expoBent of the power, which gives the third power of 2Vac is 8(ac)}=8(a?c? x ac)?=Bac(ac)? or Bacvac. 1. Raise 2(ac); to the second power. Ans. 4(ac)}. 2. Raise 4(bcx?)? to the third power. Ans. 64bcx3 bc. 3. Raise V6 to the fourth power. 4. Raise ati3 to the sixth power. Ans. a'32. 5. Raise 1+Væ to the third power. Ans. 1+31ā +3x+xvä. 6. Raise (3+2V5) to the second power. Ans. 29+125. Ans. ਤੇਠ EVOLUTION OF SURDS. 92. RULE. Extract the required root of the coefficient, and then multiply the fractional exponent of the surd by the fractional exponent of the root. EXAMPLE. Extract the square root of Vab. Here the square root of 9 is 3, and the fractional exponent of the surd is }, which we are to multiply by į the exponent of the root, which gives £; hence the quantity sought is 3(ab). 1. Extract the square root of 9/3. Ans. 393 2. Extract the square root of 36/2. Ans. 672 3. Extract the cube root of 8/5. Ans. 27 4. Extract the cube root of 27-/7. Ans. 3 x76. 5. Extract the fourth root of 643/7. Ans. 2 x (256) i' EQUATIONS CONTAINING SURDS, ETC. 93. In equations containing surds, before the solution can be effected, the surds must be cleared away; to effect this, transpose all the terms which do not contain surds to one side of the equation, and the surds to the other, then raise both sides to a power denoted by the index of the surd, and if there was only one term containing a surd, the surd will be cleared away, if there be more than one, the operation must be repeated. If an equation appear under the form xŁadã=b, or xanac"=c, it can be solved as an adfected quadratic, by solving first for the power in the second term, and then for the quantity itself. EXAMPLE. Given wx+9=Vã+1. Squaring both we have «+9=x+2Vx+1. By transposition 2V x=8. .:V=4. And squaring EXAMPLE 2. Given x + x2=72, to find the value of x. Here the equation comes under the form xen +x"=c, since the exponent of the first term is double its power in the second ; hence we must solve for x The operation will be as follows: 23 + x2=72. x=16. 3 20-43 +=2", by completing square. +1=+ by extracting the root. 28 or-9, by transposition. and 23=64 or 81, by squaring. hence x=4 or 3:33, by extracting the cube root. 17 2 EXERCISES. 3 21 1 13 1. Given v3x+435, to find x. Ans, x=7. 2. Given v4+52=2+230; to find #. Ans. x=12. 3. Given 3/20+10+4=8; to find x. Ans. x=27. 4. Given 12+ +5=7; to find x. Ans. x6. 5. Given 4x + 17+672+2=8/x+3; to find x. Ans. Q=16. 6. Given Vätv7 = ; to findx. Ans.x=16. Nx8x+4 7. Given =475+ä; to find a. Ans. 2=4. N 5+x 8. Given v7++ -; to find a Na Ans. x=9 or -16. 9. Given vite +=(1+3) i to find a. Ans. x=8. 2a 10. Given Vã+7.+a= to find X. Ans. x= Nata' 3 11. Given Vera=Nax, to find a. Ans. x= ( 12 12. Given Vx+a+va-c3b, to find x Ans. x= 5 (40—62)?! 13. Given 11+xv x2 +12=1+0, to find x. Ans.w=2. 40 14. Given vä+v16+x= to find x. Ans. x=9. 116+ ✓ +28 15. Given Vx+38 to find x. Ans. x=4. Næ+4 Nx+6 NtvT 16. Given to find a. Ans. x= varī 7 N6x-2 460-9 17. Given to find x. Ans. x=6. W 6x+2 476476 a |