8.755.000.000 | 2061 07.55 12 8000 7550.00 1200 87418 16 131840:00 127308 87545529 Si 4470 19 For reasons already explained, this number is separated into divisions of three figures each, passing from right to left. The cube root of the last division 8 is extracted; it is 2, which is written in the root. We cube 2 and subtract the product from 8, and 0 remains. By the side of 0, we bring down the period 755, of which we sepa rate the two last figures 55. Under the remaining part 7, we write 12, three times the square of the root; and dividing 7 by 12, find 0 for quotient, which is written in the root. We cube the root 20, which gives 8000 and subtract it from 8755, and have 755 for remainder. By the side of this remainder we bring down the division 000, and separate the two figures on the right; under the remaining part 7550 we write 1200, which is 3 times the square of the root 20; and dividing 7550 by 1200, find 6 for quotient, which is written in the root. We cube the root 206 and subtract the product from 8755000 and have 13184 for remainder, by the side of which we bring down the last division 000, and separate the two last figures of it. Under 131840, the remaining part, we write 127308, three times the square of 206, the root found. We divide 131840 by 127508, and find 1 for quotient, which is written to the right of 206. We cube 2061, and having subtracted 8754552981, the product, from 8755000000, have 447019 for remainder. The approximate cube root of 8755000000 is 2061; then that of 8755,000000, is 20,61; because the cube contains 3 times as many decimals as its root (54). If we wish to extend the approximation still further, three ciphers are put after the remainder, and the same operation is performed as at each time when a division is brought down. 157. Since, when a fraction is multiplied by a fraction, the numerator is multiplied by the numerator, and the denominator by the denominator; therefore, when a fraction is to be cubed, the numerator and the denominator must each be cubed. And reciprocally, when the cube root of a fraction is to be extracted, we must extract the cube root of the numerator and the cube root of the denominator. Thus, the cube root of 21 is, because the cube root of 27 is 3, and that of 64, is 4. 158. But if the denominator only is a cube, we extract the approximate root of the numerator, and give to this root for denominator, the cube root of the denominator. For example, if the cube root of 143 is required, as the numerator is not a cube, we extract its approximate root, which to two places of decimals, is 5,22; and extracting the cube root of $43, which is 7, we have 22 for the approximate cube root of 143; or by reducing this fraction to decimals (99), we have 0,74 for the root approximated to two places of decimals. tor; 159. If the denominator is not a cube, both terms of the fraction are multiplied by the square of the denominathe new denominator being now a cube, the operation is conducted as before. For example, if the cube root of is required, we multiply the numerator and denominator by 4 the square of the denominator; the fraction is now 147, which has the same value as (88). The cube root of is 5.27; or, by reducing it to decimals, 0,75; the cube root of is then 0,75, approximated to two decimal figures. If whole numbers are joined to fractions, they are converted into a fruction; the question is then to extract the 'cube root of a fraction (157 and following). The given fraction may also be reduced to decimals, whether it be accompanied by whole numbers or not; but care must be taken to extend the reduction to three times as many decimal figures as are required in the root. Thus, if the cube root of 7 is required in three places of decimals, we change the fraction into 0,2727 27 272; hence, to obtain the cube root of 73, we extract the cube root of 7,272727272, which is 1,937. 160. When the cube root is to be extracted from a number, which contains decimals, it must be prepared by placing after it such a number of ciphers as shall make the number of decimal figures either 3, 6, or 9, &c. The cube root is then extracted in the same manner as if no decimals were in the given sum; but after the operation has been performed, we separate, on the right of the root, as many figures for decimals as shall be equal to one third of the number of decimal figures in the prepared sum. If the root found does not contain so many figures as the rule requires, the deficiency must be supplied by placing ciphers to the left of the root. Thus, to extract the cube root of 6,54 to three places of decimals, we place after it 7 ciphers, and extract the cube root of 6540000000, which is 1870. To the right of this root we separate 3 figures, because there were 9 decimals in the cube, and have 1,870, or only 1,87 for the cube root of 6,54. In like manner, we find that the cube root of 0,0006, approximated to two places of decimals, is 0,08. 161. When the four first figures of the cube root have been found by the rule already explained, others may be obtained with greater facility by division, in the following manner. If the cube root of 5264627832723456 is required, we seek the four first figures of the root in the usual manner; they are 1739, and 5681413 is the re mainder, after the operation. By the side of this remainder we bring down 72, the two figures which follow 5264627832, the part which gave the four first figures of the root. (We bring down the three figures which follow this part when the root found is expressed by five figures, and four, when it is expressed by six). We divide 568141372 by 9072363, three times the square of the root 1739, and have 62 for quotient, which are to be placed to the right of 1739 as two new figures of the root. And 173962 is the cube root of the given sum expressed in whole numbers. N If the operation is to be continued further, we cube this root, and having subtracted the product from the given number, place four ciphers after the remainder, and divide the whole by three times the square of 173962; which gives four decimals for the root. EXAMPLES FOR PRACTICE. Ans. 103. 1. What is the cube root of 1092727 ? 3. What is the cube root of ,0001557 ? Ans. 3002. Ans. 05158, &c. Ans. ,67, &c. Ans. ,873, &c. 2150,4197, &c. cubic 1. What is the be root of 1520 ? 3 inches; what is the side of a cubic box which shall contain the same quantity ? Ans. 12,907, &c. inches. 7. A cube of silver, whose side is 2 inches is worth 20 dollars; what will be the side of a cube of silver whose value shall be 8 times as much? Ans. 4 inches. NOTE. To solve such questions as the preceding, cube the given side, and multiply it by the proportion between the given and required cube, and the cube root of the product will be the required side. 8. There is a cubical vessel whose side is 4 feet; what will be the side of another cubical vessel that shall contain 4 times as much ? Ans. 6,349, &c. feet. 9. A cooper, who has a cask 40 inches long, and S2 inches at the bung diameter, is required to make another cask of the same shape, but which shall hold just twice as much; what will be the length of the new cask, and its bung diameter ? Ans. 50,3, &c. inches length; and 40,3, &c. in. bung diam. 10. What is the cube root of 12,977875? Ans. 2,35. 11. What is the cube root of 36155,027576? Ans. 35,06, &c. 12. What is the cube root of ,001906624 ? 13. What is the cube root of 1219? 14. What is the cube root of 31343? Ans.,124. Ans. 2,3, &c. Ans. 3,1. 15. What is the cube root of 7}? 6 Ans. 1,93, &c. Ans. 2,092, &c. 17. If a block of marble be 47 inches long, 47 inches broad, and 47 inches deep, how many cubical inches does it contain ? Ans. 103823. 18. A cellar was dug 12 feet long, 12 feet broad, and - 12 feet deep; how many solid feet of earth were taken out of it ? Ans. 1728. 19. How many cubes of 3 inches are contained in a cubical foot? Ans 64. 20. A certain stone of a cubical form contains 474552 solid inches; what is the superficial contents of one of its sides? Ans. 6084 inches. 21. What are the two mean proportionals between 6 and 162 ? Ans. 18 and 54. NOTE. To solve such questions as the last, divide the greater extreme by the less, and the cube rcot of the quotient multiplied by the less extreme gives the less mean; multiply the same cube root by the less mean, and the product will be the greater mean proportional. 22. What are the two mean proportionals between 4 and 108 ? Ans. 12 and 36. Ans. 73. 23. What is the cube root of 389017? Ans. 638. Ans. 2805. 27. What is the cube root of ,001906624 Ans.,124. 28. What is the cube root of 219365327791 P Ans. 6031. 29. What is the cube root of 15926,972504 ? Ans. 25,16, &c. 30. What is the cube root of 673373097125 ? Ans. 8765. RULE FOR EXTRACTING THE ROOTS OF POWERS IN GENERAL. Prepare the given number by separating it into divisions if as many figures as the root required directs. |