2. Reduce 31 and 54 to surds of the same index. and / į, reduced to a common denominator, arc 3 and . 6 Now 38=33=781 ; and 58=35*=125. Ex. 3. Reduce a at Ans. Va and þes to Surds Ex. 4. Reduce ct and di with the Ans. $cs and 4d. do Ex. 5. Reduce 3 32&2 75 Ans. 3 4 & 2 7125. index. Ex. 6. Reduce 41 and 15 Ans. „256 & 73375. and 15 ) 6 same 6 6 CASE III. (131.) Surd quantities are reduced to their simplest form, by observing whether the quantity under the radical sign contains a power corresponding to the given surd root, and then extracting that root. Note.—The quantity without the radical sign is called the co-efficient of the surd; and it is evident, that this quantity may always be put under the radi. cal sigo, by rgieinw it on the power denoted by the index of the ***** Thus, 7a 2x=(by Case I.) v 7a 7a X v2.x. 49ao x 2x=98aPr. Also, -x=wX* X v2a-. => x* X (2a- x)=v2ax?. CASE IV. (132.) If the quantity under the radical sign be a fraction, it may be reduced to an integral form by the following Rule.—Multiply the numerator and the denominator of the fraction by such a quantity as will make the denominator a complete power, corresponding to the root ; then extract the root of the fraction whose numerator and denominator are complete powers, and take it from under the radical sign. Examples. The Fundamental Rules of Arithmetic applied to Surd Quantities. (133.) The Addition and Subtraction of Surd Quantities. Rule.-Reduce the quantities to their simplest form; and if the surd part be the same in both, then their sum or difference will be found by taking the sum or difference of their co-efficients, and annexing the common surd to the result. Eramples. 1. Find the sum and difference of 16aor and 74a®z. By Case III. 16aRx = 4a xx, and ✓ 4aor = 2a vX; .. the sum =4a vr+ 2a vr=(4a+2a) x vr=ba vx. the difference =40wx-2a7r=(42–2a) * vx=2a w*. 2. Find the sum and difference of 3/ 192 and 3/24. and 3 24=38 X 3 = 233; -and 3. Find the sum and difference of a Vā and 2, reduced to a common denominator, 8 The two fractions 27 48 27 are and 162 102 Now, V=V10*3=V VH-VX-V and Hence 2 ....V128 Nate.--If the surd part is not the same in the qnantities which are to be added or subtracted from each other, it is evident that the addition or between subtraction can only be performed by placing the sigus + or them. 4. Add ✓27a*x and v3a*x together.. Ans. 4ao 3r. 5. and w72 ... 1472. 6. ... 3135 and 3/ 40 535 3 7 7. Subtract 3 from 4 ✓15. 15 8. .... 108 from 934...... .634. 9. Required the sum of 2 48 and 93 108.... Ans. 8/3+27 3/4. 10. Find the difference of {v} and in Ans. 0. 11. Required the difference of „12xʻye and ✓27y*. Ans. (2xy +3y)../3. 7 (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. Examples. Qiw 6 1. Multiply va by lb, or aí by vš. The fractions and , reduced to common denominators, are 2 and õi :: ai =að= Va'; and bí=vå=162 . Hence vax a/b=ľax =Y2*6. 2. Multiply 575 by 3 8. 5/5x3/8=15/40=154 X 10. =15X2 X v10=30x10. 3. Multiply 2V/3 by 3 34. By reduction, 2/3=2 X 36=2X and 334=3 X 43=3 y 34=3716 Hence 2 13 x 3 34=3 727X3716=6432 4. Divide 23 lc by 3 vac 27bc and 3 vac=3 x (ac)ă =3 Na'co; 23 bc_2 6 / 12c2 2 6 / 19 3 ac 3 a3c3=3 aic 5. Divide 103 108 by 5 3 4. Now 103 108=10* 27 *4=13X3 X 34=3034; 107 108 30 34 103108 = 6; or 2327=2x3=6 534 534 534 6. Multiply 315 by ✓10 Ans. 225000 Х (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. Examples. 1. The square of Yaza;*2=0=80%. 2. The cube of 'b=b*3 = =6. 3. 4th power of 2.72=16x2; *4=16x2; =163 16=32 32. 4. Square root of a; b) =a} * 6*1 = 1 5. Cube root of va= x2 *!= x2 = 2 6. Find the 4th power of į v2...... .... Ans. Per 7. Find the square of 72–75...... 8 Square 35.. ..Ans. 7-2_10. Ans. 3/25. |