b y Ex. 4. Reduce and a to integral surds in their most simple form. 2/2 Ex. 5. Reduce and √ to integral surds in their most simple form. Ans. /27 and √2. to their most simple form. 3 3 28 α Ans. 2 and 2a. 4x2 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 ARITHME TIC 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 4x, x, and 5x 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=10x the sum required. Ex. 2. Find the sum and difference of 16a2x and 4a2x. (Art. 313), 16a2=✓16a2× √x=4α √x, and 4x4a3× √x=2a√x; ..the sum= =(4a+2a) × √√/x=6α √ x ; and the difference=(4a—2a) × √ x=2·1√√x. Ex. 3. Find the sum and difference of 2/108 and 93/32. Here /1083/27×3/4=3×3/4=3/4, and 93/32=93/8X3/4=18X3/4=183/4, the sum (18+3)×3/4=213/4; and the difference (18-3) 3/4=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 +er between them. Ex. 4. Find the sum and difference of 33/ab and bcd. a3b33/3×3/b=3a3/b=3a3/b, be2x√d=bc x√d=bc/d; Here the sum 3a3/b+bcd; and the difference 3u3/bbc/d. Ex. 5. Find the sum and difference of and ✓. Ans. The sum6, and difference 6. 3a1x. Ex. 6. Find the sum and difference of 27ax and Ans. The sum= (2x2+3a)√b, and difference (22°03a) ✔b. 135. 6 6 Ex. 8. Required the sum and difference of 33/625 and 23/ Ans. The sum 213/5, and difference=93/5. Ex. 9. Required the sum and difference of ab2 ̄and 3⁄4x3 y3. Ans. The sum a√ab+x3⁄4x2y2, and difference a✓abx 3x22. CASE II. To multiply or divide Surd Quantities. RULE. 343. Reduce them to equivalent ones of the same deno mination, 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. Ex. 1. Multiply a by /b, or a by bs. The fractions are and 2. and, reduced to common denominators, 3 = ..aaa3; and b3=b¤=q/b2. Hence/aX/b=/a3×/b2=/a3b3. α Ex. 2. Multiply 2/3 by 33/4. By reduction, 2/3=2× 3a2=2 × 2/33=24/27; 2 and 33/4=3X4☎=3€/42=39/16. Ex. 3. Divide 83/512 by 43/2. Here 842, and 3/512/23/256=43/4. Ex. 4. Divide 23/bc by 3/ac. Now 23/bc=2×(bc)3=2×{bc)ễ =2;/b2c2, and 3 ac 3×(ac)=3X(ac)=3%/a3c3 ; 23/be 2 6 b2c2 26 3√ ac 3 c5. 3 b2 26 b2a3c5 2 6 3ac 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 / by /a2 or as by a3. Here a3×a3=a}+}=a3==a2. Or 3⁄4/a1×3/a2=3/ {aa× a2) 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/3-2/a2 by 3+. Product /243-3/(3a2+/27a2-a. Ex. 8. Divide b3ca + √ a2b—bc - ✔abc by √bc+✔a. ✓b3ca+ub-be-abe | bc+va Quot.ba✓bc. Ex. 9. Multiply 3/15 by d. Ex. 18. Divide (a1+63)3′ by (a^+b3) 3.' Ex. 19. Multiply 4+2/2 by 2-√2. Ex. 20. Multiply √(a~√ (b−√3)) by √3)). Ans. /225000. Ans. 2/10. Ans. /ab. Ans. att Ans. a2b3c4. Ans. (a+b3). Ans. 4. √(a + √ (b-~ Ans. (d-b+√3). Ex. 21. Divide a3b-ab2c by a2+α√/bc. Ex. 22. Divide aa+xa by a2+ax√√2+x2. Ans. ab-bbc. Ans. a3-ax/2 + x2. 346. It is proper to observe, since the powers and roots of quantities may be expressed by negative exponents (Arts. 86, 311), 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. 1 Ex. 1. Thus, (since —=6-3), may be expressed by a2 Ex. 5. Let x22as be expressed with a negative exponent. To involve or raise Surd Quantities to any power. RULE, 347. Involve the rational part into the proposed power, then multiply the fractional exponents of the surd part by the index of that power, and annex it to the power of the rational part, and the result will be the power required. Compound surds are involved as integers, observing the rule of multiplication of simple radical quantities. Ex. 1. What is the square of 2/a? The square of 2/a=(2a)2=22×a112=4a. 1.3 The cube of (a2b2+√/3)=(u2—b2+√/3)3 ̈®=a2—b2+ 3. 348. Cor. Hence, if quantities are to be involved to a pow er denoted by the index of the surd root, the power required is formed by taking away the radical sign, as has been already observed (Art. 326). Ex. 3. What is the cube of 2αx. Here (1)2=}, and (√2ax)3=(2ax)¥3 =(2ax) ax/2ax is the power required. Ex. 4. It is required to find the square of ✔ub. |