2 1 28. Reduce the radical part of to a whole number. 3 3 First, the numerator 1 being multiplied by its denominator 3, 2 produces 3 for the surd; then, dividing the coefficient by 3, 3 3 20 to a whole number. Multiplying the numerator 4 by 5, gives 20 for the surd; then 3 20 whence 20 is the surd, which reduced to its simplest terms also by 5, gives for the coefficient; Multiply 2a by (the square of 5x or) 25 x2, and the product 50 ax will be the surd part; then divide the coefficient 1 (under RULE I. Reduce the quantities to a common index, the 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 =√4×2=2√2 And by adding... 5/2 the sum required. 2. Add 3/24 a, 3/192 a, and 3/81 a together. འ These reduced first to a common denominator, and then to their simplest terms, become 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 6xz. 6. Add 3/32, 3/500, and 23/4 together. Sum 93/4. 3 7. Add3/8 x3y' and 3 √16x3y' together. Sum 2x3 √y2+ 2x3√2y3. 8. Add 35/64 a3z — 32 aoz2 and 25/64 a3z—32 aoz2 together. Sum 10a3/2 z—az2. 9. Add 2+/x6, 34x3, and 4x5 together. Sum 2x√x2+ 3* √x3 +4x1 √/x. 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=3x2√7=6√/7. and 63 9x7=3√7 their difference=3√7 the Answer. 2. From 4/128 at take 2/16 a* Thus 43/128a=43 √64 a3 × 2 a=16 a3 √2 a 23 √16 a1=23 √ 8 a3 × 2 a≈ 4a3 √2 a 2 27 3. From ✔ subtract√50 Diff. 12a32a 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. x3 3 √18. 9. From 3/189 x y +27 x take 456x3y+8x3. Diff. 67. MULTIPLICATION OF SURDS. RULE I. Reduce the surds to the same index, (if they require it,) by Art. 61. then multiply the coefficients together for the rational part of the product. II. Multiply the surd parts together, and having placed the radical sign over the product, subjoin it to the former product, and 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 example 1 be proposed, where 4/2 is to be multiplied by 5/8; let the coefficients be put under the radical sign, (Art. 60.) and these quantities be come 32 and 200; wherefore 32 × √200= √✓✓✅6400=80, as in the example. Again, ex. 2, where 56 is to be multiplied by 4√3, proceeding as before, we have 150 × 48= √7200= √3600 × 2 (Art. 63.)=60/2, as in 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 5/6x4√/3=20/18=20√9 × 2=20×3√2=60 √2, the product. 4. Multiply 2+ √/⁄3 a and 3— √2 y together. Thus 2+ √3 a 6+3/3a-2/2y-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}}, and x+y=x+y'z= (x + y) 2 } ¿ • Wherefore (x+y) 3} } × (x + y) ' l z=(x+y)}}= 3 2 x3 +5x*y+I0x3y2+10x2 y3 +5 xy*+x3 \†, the product required. Prod. 6/30. 6. Multiply 2/8 by 36. Prod. 24 √3. Prod. 8 a3 √ay. 11. Multiply a+ √z and a— /z together. Prod, a2-z. 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 |