Ex. 9. Reduce (a+b)3/[(a−b)3 X x2] to its most simple form. Ans. (a2-b23/)x2 338. 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 denominator are complete powers, and take it from under the radical sign. vb. Ex. 2. Reduce to an integral surd in its simplest form. Here, IV+=+ 2X32 33 •=23/(4 × 18)=7×12/18=2/18. Ex. 3. Reduce to an integral surd in its most simple form. b Ans. 14. Ex. 4. Reduce ✓ and a to integral surds y α 339. 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√√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 processes; and the labour would be much greater for the cube and other higher roots. 340. 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 ARITHMETIC TO SURD QUANTITIES. CASE I. To add or subtract Surd Quantities.. RULE. 341. 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 4/x, x, and 5x together. √x, 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√x=10√ sum required. the Ex. 2. Find the sum and difference of √16a2 x and √4a3x. (Art. 313), 16a2x=√16a2 X√x=4α√√x, = and 4a2x=√4a2 X √x=2α√x; .. the sum (4a+2a)× √x=6α√√x; and the difference (4a-2a) Xx=2αx. Ex. 3. Find the sum and difference of 108 and 92/32. Here 3/108=3/27 x 3/4=3x/4=33/4, and 93/32=93/8×3/4=18×3/4=183/4, the sum (18+3)x3/4=213/4; and the difference (18-3)x3/4=153/4. 342. 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 (Art. 315), that the addition or subtraction of such quantities can only be indicated by placing the signs + or between them. Ex. 4. Find the sum and difference of 33⁄4/a3b and byc3d. Here a3b=33/a3×3/b=3ax/b=3a3/b, and b√✓c2d=b√✓c2 ×√/d=bc × √/d=be/d; the sum 3a3/b+bcd; and the difference 3a3/bbc/d. Ans. The sum =6, and difference and. 6. 27ar and 2a23x. ab and Ex. 8. Required the sum and difference of 33/625 and 23/135. Ans. The sum=213/5, and difference=93/5. Ex. 9. Required the sum and difference of ab2 and/xy. Ans. The sum aab+x2x2y3, and difference: a√ɑbox3/x3y3. CASE II. To multiply or divideSur d Quantities. RULE. 343. 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. I I Ex. 1. Multiply a by 3/b, or a2 by b3. The fractions, and, reduced to common denominators, are and 2. 3 I ‚'‚ a2=a* =3⁄4/a3 ; and b3=b=/b2. Hence vax/b=/a3 ×3⁄4/b2 =3⁄4/a3b3· Ex. 2. Multiply 2/3 by 33/4. 3 By reduction, 2/3=2×3a = 2 × 2/33 =22/27; and 33/4=3×4° =3°/42=3/16. Ex. 3. Divide 83/512 by 43/2. Here &÷42, and 3/512/23/2564/4. ...83/5124/2=2×43/4=83/4. Ex. 4. Divide 23/bc by 3/ac. I Now 23/bc=2× (bc)3 = 2 × (bc)3 =2;/b2c2, and 3/ac=3× (ac) = 3 × (ac)=3°/a3c3 ; 23/bc 2 6 3/ac 3 2 6 162 2 6 6 344. 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 (Art. 319, 320). Ex. 5. Multiply a by a2, or a by a3. Here 2 4 /a4 6 at+a=a+3=aš=a2. Or /aa ×3⁄4/a2 = 3/(a1 Xa3)=3/a°=a2, as before. Ex. 6. Divide a3 by 3⁄4/aa, or a 3 4 Here aaaa2 =α α by aŝ. 345. 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—3/a2 by 3⁄4/3+Va. √3a2 Since 3×2/3=38 x 38= V/3+V/a ) †(33 × 32) = √(27×9)= -6243 /243—3/(3a2) Product=243-3/3a2+/27a3—a. Ex. 8. Divide baca + a2bbc abc by ✓bc+va. √b2ca+✔a2b―bc-✔abc | √bc+√a Quot.=✓babc. -bc-abc -bc-abc |