Find the cube root of— 4. 32768. 5. 79507. 6. 157464. 7. 274625. 8. 438976. 27, T. D. 54, 1st C. 36, 2d C. 3888 SHORT METHOD OF CUBE ROOT. 229. The following Short Method of cube root is the shortest and most convenient method yet used. The abbreviation consists in obtaining the successive trial divisors by using the previous work. It is readily explained either by the blocks or by the algebraic formula. 19. Find the cube root of 48228544. PROCESS. 3276, C. D. 19656 36 432 16 393136 27 21228 48/228′544|364, Ans. 1572544 9. 614125. 10. 804357. 11. 1860867. 12. 12326391. 13. 130323843. 1572544 14. 34328125. 15. 145531576. 16. 264609288. 17. 354894912. 18. 1879080904. We find the number of figures in the root and the first and second figures of the root as before. In finding the second trial divisor, 3888, we add the first correction (54), the second correction (36), the complete divisor (3276), and the second correction (36) repeated, as indicated by the bracket in the process, and proceed as before. It is not necessary to give a full rule for this method, as it is merely a modification of the previous method. It will be easily remembered by means of the following statement: Find the first trial divisor by the usual method. The second trial divisor is obtained by adding to the first complete divisor the two corrections required to form it, together with the square of the second term. This method holds good for any trial divisor, and will be found to save a great deal of labor in extracting the cube root. .13 = .001. 27. 12895213625. 28. 160279981568. 29. 428490531819. .01.000001. .993 We thus see that the cube of a decimal contains three times as many decimal places as the decimal itself; hence to extract the cube root of a decimal we point off the decimal into periods of three figures each, beginning at the decimal point. The number of periods to the right of the decimal point shows the number of places in the decimal portion of the root. The method of ART. 229 may be used to find the cube root of a decimal if the decimal point is inserted in its proper place in the root. 30. Find the cube root of .614125. PROCESS. .614/125.85, Ans. 3 21 192 Find the cube root of 31. .079507. 32. .166375. 33. .300763. 49 559 49 120 25 20425 102125 86700 CUBE ROOT OF IMPERFECT CUBES. 231. When a number is an imperfect cube, periods of ciphers may be annexed and the process continued as far as desired. 40. Find the cube root of 5. 4590 81 8715981 512 102125 34. .636056. PROCESS. 5.000/000/000 | 1.709+, Ans. 1 4000 3913 87000000 78443829 8556171 Find the cube root of— 41. 3. 37. .000046656. 38. .000970299. 39. 36.926037. CUBE ROOT OF FRACTIONS. 232.-47. Find the cube root of }. V 7 1.912+ 8 2 V8 = 48. Find the cube root of 2. 3 36 6 √√3 = √√27 19683 166375 If the denominator is a perfect cube and the numerator is not, divide the approximate cube root of the numerator by the cube root of the denominator. = 3 Find the cube root of 49. 1728 15625 50. 27 = .956+, 1.817 + 3 51. . 52.5. Ans. If the denominator is not a perfect cube, reduce the fraction to an equivalent fraction having a perfect cube for a denominator. 53. 2. 54. . APPLICATIONS OF CUBE ROOT. 233. The Cube. The Edge of a cube is equal to the cube root of its contents. WRITTEN EXERCISES. 1. Find the edge of a cubical box which contains 216000 cubic inches. 2. Find the entire surface of a cubical chest whose contents are 15625 cubic inches. 3. What is the edge of a cube which contains as much as a solid 7 ft. long, 42 in. wide, and 21 in. high? 4. A miller has a cubical box with which he takes toll; what are its dimensions, if it contains of a bushel? 5. What are the dimensions of a cubical box that will hold 20 bushels? 6. What is the depth of a cubical cistern that will hold 1000 gallons of water? 7. B has a cubical bin whose contents are 175616 cubic inches; what will it cost to line the bottom and sides, at 10 cents a square foot? 234. Similar Volumes. Similar Volumes are those which have the same form, but differ in volume. It is proved in Geometry that 1. Similar volumes are to each other as the cubes of their like dimensions. 2. The like dimensions of similar volumes are to each other as the cube roots of their volumes. These principles may be illustrated thus: : 33. vol. MNOPQRA B': M N3. 63 : 216 27 3 √ Vol. A B C D E F : vol. M N O P Q R = AB: MN. V216 √27 = 6 3. = |