Ex. 9. Reduce (a+b)/[(a - b)3×x2] to its most simple Ans. (a2-b2)/x. form. 275. If the quantity under the radical sign be a fraction, it may be reduced to a whole quantity, thus: Multiply both the numerator and denominator 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 den minator are complete powers, and take it from under the radical sign. Ex. 3. Reduce √ to an integral surd in its most simple form. b Ans./14. Ex. 4. Reduce - and a3/- to integral surds in their most simple form. y a Ex. 5. Reduce √ and to integral surds in their most simple form. Ans. 27 and 2. a3 8x4 276. The utility of reducing surds to their most simple forms, especially when the surd part is fractional, will be readily perceived from the 3d example above given, where it is found that 3√3=2/14, in which case it is only necessary to extract the square root of the whole number 14, (or to find it in some of the tables that have been calculated for that purpose), and then multiply it by; whereas we must, otherwise, have first divided the numerator by the denominator, and then have found the root of the quotient, for the surd part; or else have determined the root of both the numerator and denominator, and then divide the one by the other; which are each of them troublesome pro cesses; and the labour would be much greater for the cube and other higher roots. 277. There are other cases of reducing algebraic Surds to simpler forms, that are practised on several occasions; for instance, to reduce a fraction whose denominator is irrational, to another that shall have a rational denominator. But, as this kind of reduction requires some farther elucidation, it shall be treated of in one of the following sections. § III. APPLICATION OF THE FUNDAMENTAL RULES OF ARITH METIC TO SURD QUANTITIES. CASE I. To add or subtract Surd Quantities, RULE. 278. Reduce the radical parts to their simplest terms, as in the last case of the preceding section; then, if they are similar, annex the common surd part to the sum, or difference of the rational parts, and it will give the sum, or difference required. Ex. 1. Add 4x, x, and 5/ together. Here the radical parts are already in their simplest terms, and the surd part the same in each of them; .. 4√x+√x +5x=(4+1+5)× √√√x=10√√√r the sum required. Ex. 2. Find the sum and difference of /16ar and 4ax. 16a2x=16a2 Xx=4α, and 4a4a2 Xx=2a; .. the sum (4a+2a) Xx=6α/x; = and the difference (4a-2a) Xx=2α1/x. Ex. 3. Find the sum and difference of 3/108 and 93/32. Here/108/27X3/4=3X3/4=33/4, and 9/32=93/8X3/4-18X3/4=18/4, the sum (18+3)×3/4=213/4; and the difference (18-3)x3/4=153/4. 279. If the surd part be not the same in each of the quantities, after having reduced the radical parts to their simplest terms, it is evident that the addition or subtraction of such quantities can only be indicated by placing the signs between them. or Ex. 4. Find the sum and difference of 33/ab and b/cd. Here 33/ab 33/a3×3/b=3a3/b=3a3/b, and b/c db/c2×/d=bcx/d=bcλ/d; and the difference=3a3/bbcd. Ex. 5. Find the sum and difference of and /. Ex. 7. Find the Ans. The sum = Nb. 2 3a +34) √b, and difference (222-3α) 2x2+3a Ex. 8. Required the sum and difference of 33/625 and 23/135. 3 Ans. The sum =213/5, and difference =93/5. Ex. 9. Required the sum and difference of ab' and 3⁄4r3y. Ans. The sum a1ab+x2x22, and difference a1ab CASE II. To multiply or divide Surd Quantities. RULE. 280. Reduce them to equivalent ones of the same denomination, and then multiply or divide both the rational and the irrational parts by each other respectively. The product or quotient of the irrational parts may be reduced to the most simple form, by the last case in the preceding section. or at by b3. Ex. 1. Multiply a by /b, or a by The fractions and 3, reduced to a common denominator, are and . 6 :-.a* =a*='/a3; and b3—b¤—§/b2. Hence /ax/b=v/a3×v/b2=v/a3b3. Ex. 2. Multiply 2/3 by 33/4. By reduction, 2/3=2×36=2X/33=25/27 ; and 33/4=3x4=3/42=3/16. • 2/3×33/4=25/27×35/16=6/432. Ex. 3. Divide 83/512 by 43/2. Here 842, and 3/512/2=3/256=43/4. Ex. 4. Divide 23/bc by 3/ac. Now 23/bc=2x(bc)=2× (bc)* =2;/b2c2, and 31/ac=3×(ac)3 =3× (ac)*=3;/a3c3 ; 23/bc 2 6 bac2 26 ba 26 3/ac 3 3 a3c 34 3ac 281. If two surds have the same rational quantity under the radical signs, their product, or quotient, is obtained by making the sum, or difference, of the indices, the index of that quantity. 4 Ex. 5. Multiply /a by /a2 or a3 by 6 Here aaa+÷_at. =/aa2, as before. a2 by at. 12 282. If compound surds are to be multiplied, or divided, by each other, the operation is usually performed as in the multiplication, or division of compound algebraic quantities. It frequently happens that the division of compound surds can only be indicated. Ex. 7. Multiply /3-2/a2 by //3+/a. 6 3 /3-/a2 Since /3X/3=3′′ X36= -243 /243-3/(3a2) Product/243-3/(3a)+27a"-a. Ex. 8. Divide b3ca+/a2b-bc-abc by bc+va. √b2ca+a2b-bc-√abc | √bc+√a Ex. 16. Multiply a3 x* by a*z*. Ex. 17. Multiply {/a2b3c1 by */a2b3c3. Ans. a Ans. a2b3c4. Ans. (a+b3). Ans. 4. Ex. 19. Multiply 4+2/2 by 2-2. √3)). Ex. 21. Divide a3b-abc by a2+a/bc. Ex. 22. Divide a+2 by a2+ax/2+23. Ans. ab-bbc. Ans. a2-ax/2+x2. 283. It is proper to observe, since the powers and roots of quantities may be expressed by negative exponents, that any quantity may be removed from the denominator of a fraction into the numerator; and the contrary, by changing the sign of its index or exponent; which transformation is of frequent occurrence in several analytical calculations. Ex. 2. The quantity may be expressed by a2b3c—e—5. Ex. 5. Let x2ya1 be expressed with a negative exponent. |