1 fractions to a common denominator, (Art. 180. Part I.) and the surds to their simplest terms, Art. 63. II. If the surd part be alike in all the quantities, add the coefficients together, and to their sum subjoin the common surd. III. If the surd parts be unlike, the quantities cannot be added, otherwise than by connecting them together by means of their signs *. EXAMPLES. 1. Add/8 and 18 together. These reduced to their simplest terms, are, √/8 = √4x2=2√2 And by adding... 5/2 the sum required. 2. Add 3/24 a, 3/192 a, and 381 a together. Thus $√24 a=3 √ 8×3a=23 √3 a. 3/192 a 3/64 x 3 a=43/3 a. 381 a=√27 × 3a=33√3a. These reduced first to a common denominator, and then to their simplest terms, become 3 v3. This rule is sufficiently plain, without any further illustration than what is contained in the notes on addition of rational quantities, Art. 36. to 40. 4. Add 8 and 5. Add /4 x'z and 32 together. Sum 6/2. 16x'z together. Sum 6x √z. 6. Add3/32, 3/500, and 24 together. Sum 93 √4. 3 7. Add8ry' and ' 16x'y' together. Sum 2x3 √y2+ 8. Add 35/64 a3z-32 aoz' and 25/64 a3z—32 az3 together. Sum 10a5/2z—az3. 9. Add 24/6, 34x3, and 4* / together. Sum 2x√x2+ 66. SUBTRACTION OF SURDS. RULE I. Prepare the quantities (if they require it) as in addition. II. Consider which surd is to be subtracted, and, if both surd parts are alike, subtract its coefficient from the coefficient of the other, subjoining the common surd to the remainder. III. But if after the necessary reduction the surd parts are unlike, change the sign of the quantity to be subtracted, and then connect it with the other quantity ". EXAMPLES. 1. From 3/28 take √63. These reduced to their simplest terms, are, 3/28=3/4x7=3×2√7=6√/7 and √63= √9x7=3√7 their difference=3√7 the Answer. 2. From 4/128 a1 take 23/16 a* Thus 43/128 aa=43 √√⁄/64 a3 × 2 a≈ 16 a3 √2 a 23 √16 a1=23 √ 8 a3×2 a≈ 4a3 √2 a 3. From ✓ 2 3 subtract 27 50 Diff. 12 a3/2a Ans. "Subtraction of surds evidently depends on the same principles with subtraction of rational quantities, as will readily appear from a bare contemplation of the rule. 4. From 50 take 18. Diff. 2√2. 5. From 3/175 take 2/28. Diff. 11/7. 6. From 250a3x take/16 a'x. Diff. 3 a3 √2x. From3/250 7. From 8a take 8a". Diff. 2√2 a−2 a√2.. 9. From 3189xy+27 x take 456x3y+8x3. Diff. 67. MULTIPLICATION OF SURDS. RULE I. Reduce the surds to the same index, (if they require t,) by Art. 61. thên multiply the coefficients together for the ational part of the product. II. Multiply the surd parts together, and having placed the adical sign over the product, subjoin it to the former product, nd reduce the surd to its simplest terms *. Art. 63. EXAMPLES. 1. Multiply 4/2 by 58. Thus 4×5=20, the coefficient, and √2× √8=√16, the * To shew that this mode of operation agrees with known principles, let ample 1 be proposed, where 42 is to be multiplied by 5/8; let the efficients be put under the radical sign, (Art. 60.) and these quantities be me √32 and 200; wherefore √32 × √/200= √6400=80, as in the imple. Again, ex. 2, where 5/6 is to be multiplied by 4√3, proceeding as fore, we have 150 × √48= √7200= √3600 × 2 (Art. 63.) = 60 √/2, as the example, and the like may be shewn in all other cases. surd part: therefore 2016, or 20√4×4=20×4=80, the answer. 2. Multiply 5/6 by 4√√3. Thus 56×4√√3=20√✓19=20 √√/9×2=20×3√2=60 √2, the product. Thus 2+ √3 a 6+3√3a−2√2 y−√√6 ay, the product. 5. Multiply x+y+ and x+y+ together. These reduced to a common index, become x+y};=x+y}}=(x+y)3⁄43}}, and x+y÷=x+y} z=(x+y)3¿• Wherefore (x+y)3}z × (x+y)3] z=(x+y)]}= 3 x2+5x*y+10x'y' +10x'y' +5 xy+x, the product required. Prod. 6/30. 6. Multiply 28 by 36. Prod. 24/3. 1 2 6 together. Prod. ✔z together. Prod. a2 —z. 11. Multiply a+ √z and a— 68. DIVISION OF SURDS. RULE. Reduce the surds to the same index, (Art. 61.) if they require it, then divide the rational parts by the rational, and the surd by the surd; the former quotient annexed to the latter will be the quotient, which must be reduced to its simplest terms as before.y. EXAMPLES. 1. Divide 20/21 by 4/3. 20 Thus -=5 the rational part of the quotient. And 4 123/48 √2 =2' √/24=23 √/8x3=43√3, the quotient. 1 Thus 63 4. Divide x-x2zz by x-x √z. x-x/z) x2-x2z √√/z (x+x√/z+xz. Quotient. 2 x2x2 No7 √9×83= √4608= 4 y Division being the converse of multiplication, its method of operation, which is manifestly the converse of that of multiplication, must needs be true; but it may likewise be shewn to be so in a similar manner to that employed in the preceding note; thus ex. 1. where 20/21 is required to be divided by 43; putting the coefficients of both under the radical sign, (Art. 60.) and √8400 dividing the former by the latter, we have √48 (Art. 63.) =57, as in the example referred to; and the same may be shewn to hold true in all other cases. |