21. Reduce 3√128x3 to its simplest form. Ans. 24x√2x Ans. 24x√/3. Ans. 2√2+3a3. Ans. 3x(a+b)/3 Ans. 3a√x-2a. 27. Reduce 3√a2x-2a3 to its simplest form. 28. Reduce (a—x)/192a3x to its simplest form. Ans. 4a(a—x)√3x. 29. Reduce 4√4x2+8x3 to its simplest form. Ans. 8x1+2x. to an integral Surd in its simplest form. (238). 36. Reduce 5√2 and 3/4 to Surds of the same root. (239). By reducing the exponents of the Surd factor to the common denominator, we find 3 2a—28—8%; and 4—4&=16*. Prefixing the rational co-efficients, we have Ans. 5/8, and 3/16. 5√/2=5/8, and 33/4=3/16. 37. Reduce 2/5 and 5/3 to Surds of the same root. 3 Ans. 2/25, and 5′27. 38. Reduce a√5 and x/2 to Surds of the same root. Ans. a/125, and 26/4. 39 Reduce 10/10 and 24/3x to Surds of the same root. Ans. 10/100, and 24/3x 10. Reduce 7/3y and 2y√xy to Surds of the same root 12 Ans. 71/81y, and 2y13y 41. Reduce a2x√2 and 3√a2x to Surds of the same root. Ans. a2x, and Va⭑x2. 43. Reduce 2x√a and to Surds of the same root. У 8x3 2x y2 1 45. Reduce and to Surds of the same root. ADDITION AND SUBTRACTION OF SURDS. (240.) 1. The Sum, or Difference, of similar surds is obtained by prefixing the sum, or difference, of their coefficients as a coefficient to the common radical factor. 2. Dissimilar surds can be added together, or subtracted the one from the other, only by the proper sign; but Surds apparently dissimilar often become similar when reduced to their simplest forms, (237). EXAMPLE. To Add together 5/80a and 3/125a. Reducing the surds to their simplest forms, we find and 31/125a=3√/25√/5a=15√/5a. The two given Surds have thus become similar, (232.) Adding together the coefficients 20 and 15, we find the Sum 35/5a. The Difference of the two given Surds, is (20-15)√5a=5√/5a. The Addition or Subtraction of similar Surds is evidently nothing more than the addition or subtraction of similar monomials; thus 20 times/5a+15 times √5a is 35 times √5a; just as 20a +15a is 35a. EXERCISES On the Addition and Subtraction of Surds. 1. Find the Sum of 3√/27 and 21/48. 2. Find the Difference between 50 and √72. 3. Find the Sum of 7√28 and 6√/63. 4. Find the Difference between 21 √2 and 5√18. 5. Find the Sum of 180 and 405. Ans. 17√3. Ans. √2. Ans. 32√7. Ans. 6√2. Ans. 151/5. 6. Find the Difference between 18 and 2/50. Ans. 7√2. Ans. 5a√3. 7. Find the Sum of √12a2 and √27a2. 8. Find the Difference between 3√/24x2 and √54x2. Ans. 3x√6 9. Find the Sum of 4√3a and √/48a. Ans. 8√3a. 10. Find the Difference between √4a3 and √9a3. 11. Find the Sum of 3/4 and 74. Ans. ava. Ans. 10/4 Ans. 78√2. Ans. 222. 12. Find the Difference between 9√200 and √288. 13 Find the Sum of 4/54 and 23/250. 3 3 14. Find the Difference between 5√/9x3 and 3√x3. Ans. 12x√x. 15. Find the Sum of 2/16a and 54a. 16. Find the Difference between 3/10 and 5/10. 17. Find the Sum of 5/98x and 10√2x. 18. Find the Difference between 3a/5 and a/5. 19. Find the Sum of avÿ2 and 363⁄4y2. 3 3 Ans. 7/2a. Ans. 45√/2x Ans. 2a/5. 3 Ans. (a+3b) Vy2. 20. Find the Difference between 5/5 and 2a√/5. Ans. (5—2a)√/5 21. Find the Sum of (a+1)3 and √/4a+4. Ans. 3 (a+1)3 22. Find the Difference between √/1+2 and 3 (1+x)3. Ans. 2(1+x)3 23. Find the Sum of 2 (a—x)3 and √/9a—9x). Ans. 5 (a-x) 24. Find the Difference between 3/2+y and 4 (y+2)3. Ans. 3 (2+y)3 25. Find the Difference between 4 (1+x2) and 4 x2+1. MULTIPLICATION AND DIVISION OF SURDS. (241.) 1. The Product, or Quotient, of two Surds of the same root, is obtained by prefixing the product, or quotient, of their coefficients as a coefficient to the product, or quotient, of the radical factors,—the latter being affected with the same fractional exponent or radical sign. 2. Surds of different roots may be reduced to equivalent ones of the same root, (239), and then multiplied, or divided, as above. But 3. Any two roots of the same quantity may be multiplied into each other, by adding together their fractional exponents; or divided, the one into the other, by subtracting the exponent of the divisor from that of the dividend. EXAMPLE. To find the Product of 2√/10×3√/2. Since it is immaterial in what order the four factors are taken, we may take them in the order; 2x310×2; which gives the Product 6√/20, (234), =12√/5, (237) The Quotient of 2√/10÷3√2 is √5, since this quotient multiplied by the divisor produces the dividend. EXERCISES On the Multiplication and Division of Surds. 1. Find the Product of 5√/8x3√/5. Ans. 30/10. Ans. 6/3. Ans. 120ax. Ans. 3ax√3 Ans. 2x√3. Ans. 109. 10. Find the Quotient of 43/72÷23/18. Ans. 234. 18×54. |