48 162 = X3 4 81 X 2 9 2 27 3 9 Now, V-V HxV Virt VH-VXV and 162 81 X2 9 3 Hence 9 2' 2 Note.--If the surd part is not the saine in the quantities which are to be added or subtracted from each other, it is evident thai the addition or subtraction can only be performed by placing the sigos + or — between them. 4. Add 27a*x and v3a*x together. Ans. 4ao „3x. 5. ✓ 128 and w72 142. 6. 3135 and 3/ 40 ..535 5 7. Subtract 3 from 4 1 V 15. . 8. .... 7 108 from 934 ...63 4. .9. Required the sum of 2 48 and 9 3 108.... Ans. 8/3+27 34. 10. Find the difference of į vį and vi Ans. 0. 11. Required the difference of „12x’ye and 27y*. Ans. (2xy +3y?).w3. (134.) The Multiplication and Division of Surd quantities. Rule.- Reduce the quantities to equivalent ones with the same index, and then multiply or divide both the rational and the irrational parts by each other respectively. 1. Multiply va by Yb, or aí by vš. The fractions and ș, reduced to common denominators, are 2 and 6} ... aí=aš= ľa; ti=8 =15x2x10=30/10. 3. Multiply 2V3 by 3 3 4. By reduction, 27/3=2x38=2x 13°=2.727, ac and 3 y 4=3 x 43=3> 48=3216 = 6/19c2 2 6 / 12 3 ac 3 F3 5. Divide 103 108 by 5 3 4. Now 10 3 108=10* 27 *4=13x3 x 34=3034; 103 108 3034 103 108 = 6; or 534534 2327=2 X3=6 534 6. Multiply 3/15 by V10 Ans. 225000 2 7. .. k yo by 18.. Ans. 34. 8. Divide 10/27 by 203 Ans. 15. 9. 103 108 by 5 3/84. Ans. 3441. 10. Multiply 5x7 by 4t. Ans. 2015. •v-y by vy..... .. 2 . 11. (135.) On the Involution and Evolution of Surd quantities. Rule.-Raise the rational part to the power or root required, and then multiply the fractional index of the surd part by the index of that power or root. X4 Examples. 1. The square of Va=a;*?==Ya'. e 2. The cube of 'v=b}*3 = 5=>0. 3. 4th power of 232=16x2; *4=16x2=163 16=3272. 4. Square root of a 0 =a; *10*1 = 0 ; 5. Cube root of kva= x2:*=x2;=72. 6. Find the 4th power of į V2.... Ans. i. 7. Find the square of 72-75.. 8 Square 35... Xiai = X2 8r Ans. 7-210. Ans. 3/2.5. 10. Cube v3.. ŽV3..... 11. Find fourth power of av... äv6....... Ans. 6 Ans. ava. 1 36. 12. Find square root of 933..... Ans. 393. 16 13. Find fourth root of 26 819a..... . 1 14. Find fifth root of $20 Х Ans. 32 2%a'. (136.) Scholium.-From the preceding rules we easily deduce the method of converting fractions, whose denominators are surd quantities, into others, whose denominators shall be rational. Thus, let a both the numerator and denominator of the fraction wä be multiplied QNI by wr, and it becomes ; and by multiplying the numerator and comes denominator of the fraction by y (a + x)* or (a +x), it be zati b(a+1)3_b(a+1)3. Or in general, if both the numerator +x) ata and denominator of a fraction of the form av de multiplied by V.co-s, ait becomes a fraction whose denominator is a rational a a quantity. On the method of finding Multipliers which shall render Binomial Surd Quantities Rational, (137.) Compound surd quantities are those which consist of two or more terms, some or all of which are irrational ; and if a quantity of this kind consist only of two terms, it is called a binomial surd. The rule for finding a multiplier which shall render a binomial surd quantity rational, is derived from observing the quotient which arises from the actual division of the numerator by the denominator of the following fractions. Thus, 1. Zelthony + som sy® + &c...tya-' to n térms, whe 1-y ther n be even or odd. 2. 3. a-o **--4°=x-1--****y+z=342 —&c.. –y– to n terms, when x+y n is an even number. 2* +y" =x-l-xampy + x-3y2– &c.. tyn-' to n terms, when x+y n is an odd number.* (138.) Now let zn=a, ya=b, then r=va, y=vo, and these rax, amb atb fractions severally become and Data+ by the application of the rules, already laid down, we have -= 7-= Pan-; ?s=303; 2*-= Van-s, &c.; also, yı=64; Par–7a ; ; yo702 y•=bo, &c. hence, zasy= Var-x 10 = Var-ab; zn–3yo = Par-sx 112=12*—36%, &c. By substituting these values of thes, x-ay, x-3y?, &c. in the several quotients we have a-6 = Va*-+ Mar-1+ van–3b2+ &c.. +16*- to n terms; 1 where n may be whole number wbatever. And ato Wat where the terms b and on- have the sign +, when n is an odd number; and the sign i -, when n is an even number. n (139.) Since the divisor multiplied by the quotient gives the dividend, it appears from the foregoing operations that if a binomial surd of the form va-vb be multiplied by van-e+ Van-3b+ var-3b2+&c.... + vb- (n being any whole number whatever), the product will be a—b, a rational quantity; and if a binomial surd of the form wat vb be multiplied by va -par-16+'van-she-&c.... +2b-, the product will be --Y 3: 3Y3 • For 1. =x+y; x+y+y^; =x3+x2y + xys +y, &c. x-y 2,13—2y+xyz-yo, &c. x+y 35+y =rl_x34+xiya-43+y4, &c. Try xty x+y 2. n a+b or a-b, according as the index n is an odd or an even number ; but here we inust observe, that the number of terms to be taken, is always equal to n. Note.—The greatest use of this rule is, to convert fractions having surd denominators, into others which shall have rational ones. Examples. Now a 1. Reduce to a fraction which shall have a rational deno. -VI minator. Vr=va? – WX; put aʻ=c, then wa’ – Nx=vc-vx. In this example n is equal to 2, and therefore the number of terms to be taken of the general series ur-!+ Man~+&c. is 2, and because a=c and l=x, the multiplier is c+ i'r; whence 13/0+ ur *(c+r)_ar+IVO = C-1 a-X 76 2 Reduce to a fraction which shall have a rational de 78+73 nominator. Here again n=2: therefore the number of the terms to be taken of the series, Pan-i-van-zb+&c. is 2; and in this example, a=8 and l=3 : therefore the multiplier is V8-73; whence V8-73 76 473-32 x 2 3. Reduce to a fraction which shall have a rational deno 33- 32 minator. Here n=3, a=3, and b=2; therefore by substituting those in the general series, Van-s+ Var=26+ Van–322+ &c.; it will become, 33 + 3/32+ 3/2=39+ 36+ 34, which is therefore the multiplier ; whence 39+ 16+ 34 2 x 29+3 6+ 34*33-32 3-2 a n 11 2(79+36+34) =2(39+36+34) с 4. Reduce to a fraction which shall have a rational deno. 3' x + yy minator. Here n=3, a=r, and b=y; therefore by substituting 3, x and y, for n, a and be in the general series, van-i-par-36+ Var-302 +&c. gives 34x2 – 3 xy + 3y for the multiplier ; whence ¥ r2- 3 xy + 3y2 c( 3 x — 34xy tyy?) ¥r — ¥xy+ Zy? ^ Jx+3y x+y 5 5. Reduce to a fraction which shall have a rational deno 47+ 3 mina'cr. с Х a |