5. Then square this last figure in the quotient, and add it to the product just mentioned, for the second part of the divisor; all these products added, form the true divisor. 6. Now multiply this true divisor by the last figure put in the quotient; subtract the product from the dividend as in division, bring down the next period of three figures; proceed as before. For Decimals.-Annex ciphers to the decimal, if necessary, so that it shall consist of 3, 6, 9, &c., places; then put the first point over the place of thousandths, the second over the place of millionths, and so on over every third place to the right; and then extract the root as in whole numbers. Observe, 1. There will be as many places in the root as there are periods in the given number. 2. The same rule applies when the given number is composed of a whole number and a decimal. 3. If there be a remainder in a whole number, after all the periods have been brought down, you can annex periods of ciphers, considering them as decimals. For Vulgar Fractions.-1. Reduce compound fractions to simple ones, mixed numbers to improper fractions, and then reduce the fraction to its lowest terms. 2. Then extract the cube root of the numerator and denominator separately, if they have exact roots ; but if either of them have not an exact root, reduce the fraction to a decimal, and extract the root as above. Proof.-Cube the root and add in the remainder. EXAMPLES. 1. What is the cube root of 5382674 ? 1x1x300=300 5382674(175.2 root. 1x7 x 30=210+ 1 [Ans 7x7 49 =559)4382 3913 446375 9198004)23299000 18396008 od Divisor, 9198004 4902992 rem. RULE II. 1. Point off the given number into periods of three figures each; if whole numbers commencing at the right hand; but if decimals, at the left. 2. Find the greatest cube in the left hand period, and place che root to the right of the given number, and subtract the cube of the root from the left hand period, and to the remainder bring down the next period for a dividend. 3. Square the root, and multiply it by three for a defective divisor. 4. Reserve mentally the units and tens of the dividend, and try how often the defective divisor is contained in the remainder; place this result to the root, and the square of it to the right of the defective divisor, but if the square is less than ten, supply the ten's place by a cipher. 5. Complete the divisor by adding thereto the product of the last figure of the root, and the remaining figure or figures of the root, and that again by 30; then divide and subtract as in long division. 6. The defective divisors after the first may be easily found by adding to the last divisor the number that completed it, with twice the square of the last figure of the root. 7. Thé divisor is then completed according to the 4th and 5th paragraphs above. 2. $/3796416 3796416(156 Square of 1 muliplied by 3 and 1 the square of 5 added, is 325 The 5 paragraph produces=150 2796 - 2375 Complete divisor=475 The number that com 421416 pleted it, 150 421416 Twice the square of the last figure of the root, 50 Square of 68, defective divisor, 67536 Paragraph 5 produces, 2700 Complete divisor, 70236 3. What is the cube root of 99252.847 ? 4. What is the cube root of 259694072 ? 5. What is the cube root of 34328125 ? 6. What is the cube root of 12.977875 ? 7. What is the cube root of 171.46776406 ? Ans. 638. 325. 2.35. 5.555+ 17 +87 rem 8 What is the cube root of .5 ? Ans. 319 3002 .0909. 1.505. 9261 1 9. What is the cube root of 32461759 ? 10. What is the cube root of 27054036008 ? 11. What is the cube root of .751089429 ? 12. What is the cube root of 3.408862625 ? 13. What is the cube root of 250 ? 14. What is the cube root of 31 15 ? 343 15. What is the cube root of 5? 16." What is the cube root of 1913 ? 17. What is the cube root of o ? 18. What is the cube root of 84.604519 ? 19. What is the cube root of {a} ? 20. What is the cube root of 16194277 ? 21. What is the cube root of 54854153 ? 22. What is the cube root of 1728 23. What is the cube root of 729.? 133T 24. What is the cube root of 7 ? 25. What is the cube root of 15625 ? 26. What is the cube root of 436036424287 ? 27. What is the cube root of 99 ? 31 .822 zi 13+ 4.39 . 253. 379.958793+ ? II 1.9129. 25 4.62+ APPLICATION. To find two mean proportionals between any two given numbers RULE. Divide the greater by the less, and extract the cube root of the quotient. 2. Multiply the root so found by the least of the given numbers, and the product will be the least. 3. Multiply this product by the same root, and it will give the greatest. 28. What are the two mean proportionals between 6 and 750 ? Thus : 750–6==125 and 125=5, then 5x6 =30, least, and 30 x 5=150 greatest; 30 and 150. Ans. Note.—The solid contents of similar figures are in proportion to each other, as the cubes of their similar sides or diaineters. 29. If a bullet 6 inches in diameter weigh 32 lb., what will a buliet of the same metal weigh, whose diameter is 3 inches ? As 6 x 6 x6=216; 3x3x3=27; as 216: 32 lb. :: 27 : 4 lb. The side of a cube being given, to find the side of that cube which shall be double, triple, &c., in quantity to the given cube. RULE. Cube your given side, and multiply it by the given propertion between the given and required cube, and the cube root of the product will be the side sought. 30. If a cube of silver whose side is 4 inches, be worth D50, I demand the side of a cube of the like silver, whose value shall be 4 times as much ? Ans. 4X4X4=64, and 64 X4=256 : 256=6.349+inches. 31. There is an oblong cellar, the content of which is 1953.125 cubic feet; what is the side of a cubical cellar that shall contain just as much ? Ans. 12.5 feet. 32. What is the difference between a solid half foot and half of a solid foot ? Ans: 3 half feet. 33. In 221184 solid inches how many cords ? Ans. 1 cord. 34. Admitting a room to be 11 feet high, 21 feet in length, and 16 feet in width, what number of cubic feet of space in it ? Ans. 3696 cubic feet. 35. The diameter of a bushel measure being 18.1 inches, and the height 8 inches, what is the side of a cubic box which shall contain that quantity ? Ans. 12.907 +inches. 36. In a cubic foot, how many cubes of 6 inches, and how many of 4, of 3, of 2, of 1, are contained therein ? Ans. 8 of 6 inches ; 27 of 4 inches ; 64 of 3 inches ; 216 of 2 inches ; 1728 of 1 inch. 37. Suppose a cubical cellar to contain 1728 solid feet, what will one of its cubic sides measure ? 38. In a square box that will contain 1000 marbles, how many will it take to reach across the bottom of the box, in a straight Ans. 10. 39. What is the difference between the cube root of 27 and the square root of 9 ? Ans. O 40. If a globe of silver 3 inches in diameter be worth D160, what is the value of one 6 inches in diameter ? Ans. 33 : 63 :: D160 : D1280. 41. If the diameter of the planet Jupiter is 12 times as much as the earth, how many globes of the earth would it take to make one as large as Jupiter ? Ans. 1728. 42. There are two small globes ; one of them is one inch in diameter, and the other two inches ; how many of the smaller globes will make one of the larger ? Ans. 8. row ? RULE III. 1. Find the quotient root of the left hand period, which subtract from the same, and then bring down the next period. 2. Multiply the square of the quotient figure by 300 for a divisor; then find the next figure ; square this quotient figure ; multiply that square by the other quotient figure, and then by 30; find the cube of this last quotient figure ; add both these products to the product of the divisor and the quotient figure, the sum of which subtract from the dividend. 3. Then bring down the next period, which will complete the next dividend ; square your two quotient "figures, and multiply by 300 for your next divisor, and so continue till the operation is completed. Note.—The roots of the 4, 6, 8, 9, and 12 powers may be obtained in the following manner : For the 4th root, take the square root of the square root. REVIEW. What is a cube? What is the cube root ? What will you first do to extract the cube root of a whole number ? Repeat the rule ? How do you extract the cube root of a decimal fraction ? When there is a decimal and whole number, how will you point them off? Why do you point decimals from the left or decimal point toward the right? In extracting a root, if there be a remainder, what may be done? What is the rule for decimals? How do you extract the cube root of a vulgar fraction? What is the rule ? How do you extract the cube root of a mixed number? What is the difference between a cube and the cube root ? 43. What is the cube root of 3636 ? Ans. 3.32+. ALLIGATION MEDIAL. ALLIGATION is used when the quantities and prices of several things are given, to find the mean price of the mixture composed of those materials ; or the method of mixing two or more simples of different qualities, so that the composition may be of a mean or middle quality. There are two kinds, Alligation : ledial and Alligation Alternate. |