divisor, (or, bring down your last divisor for a new one, doubling the right hand figure of it,) and from these find the next figure in the root, as last directed, and continue the operation in the same manner, till you have brought down all the periods. Note 1. If, when the given power is pointed off, as the power requires, the left hand period should be deficient, it must nevertheless stand as the first period. Note 2. If there be decimals in the given number, it must be pointed both ways from the place of units. If, when there are inte. gers, the first period in the decimals be deficient, it may be completed by annexing so many ciphers as the power requires. And the root must be made to consist of so many whole numbers and decimals, as there are periods belonging to each; and when the periods belonging to the given numbers are exhausted, the operation may be continued at pleasure by annexing ciphers. Note 3. If it be required to extract the square root of a vulgar fraction, reduce the fraction to its lowest terms, then extract the square root of the numerator for a new numerator, and of the denomi. nator for a new denominator ; or, reduce the vulgar fraction to a decimal, and extract its root. 2. What is the square root of 148996 ? OPERATION. 1 4 8996(386 9 544 45 96 3. What is the square root of 23804641 ? Ans. 4879. 4. What is the square root of 10673289 ? Ans. 3267. 5. What is the square root of 20894041 ? Ans. 4571. 6. What is the square root of 1014049 ? Ans. 1007. 7. What is the square root of 516961 ? Ans. 719. 8. What is the square root of 182329 ? Ans. 427. 9. What is the square root of 61723020.96 ? Ans. 7856.4. 10. What is the square root of 9754.60423716 ? Ans. 98.7654. 11. What is the square root of 3331? Ans. 61 12. What is the square root of 12345? 13. What is the square root of 9 ? 14. What is the square root of 129? 15. What is the square root of 6016 ? 16. What is the square root of 2837 ? 17. What is the square root of 4711? Ans. APPLICATION OF THE SQUARE ROOT. 18. A certain general has an army of 226576 men ; how many must he place rank and file to form them into a square ? Ans. 476. Note. In a right angle triangle, the square of the longest side is equal to the sum of the squares of the other two sides. 19. What must be the length of a ladder to reach to the top of a house 40 feet in height ; the bottom of the ladder being placed 9 feet from the sill ? Ans. 41 feet. 20. Two vessels sail from the same port; one sails due north 360 miles, and the other due east 450 miles ; what is their distance from each other ? Ans. 576.2+ miles. 21. If a pipe, 2 inches in diameter, will fill a cistern in 204 minutes, how long would it take a pipe, that is 3 inches in diameter ? Ans. 9 minutes. 22. If an anchor, which weighs 2000 lbs., requires a cable 3 inches in diameter, what should be the diameter of a cable, when the anchor weighs 4000lbs. ? Ans. 4.24+ inches. 23. How large a square stick may be hewn from a round one, which is 30 inches in diameter ? Ans. 21.2+ inches square. 24. John Snow's dwelling is 60 rods north of the meetinghouse, James Briggs is 80 rods east of the meetinghouse, Samuel Jenkins' is 70 rods south, and James Emerson's 90 rods west of the meetinghouse ; how far will Snow have to travel to visit his three neighbours, and then return home ? Ans. 428.4+ rods. Section 49. EXTRACTION OF THE CUBE ROOT. A Cube is a solid, bounded by six equal squares. A number is said to be cubed, when it is multiplied into its square: To extract the cube root, is to find a number, which, being multiplied into its square, will produce the given number. The extraction of this root has been illustrated by mathematicians in various ways. But it is believed, that · Robert Record, Esquire, of London, in his Arithmetic published in 1673, was among the first, who illustrated this rule by the use of various diagrams and blocks. The same thing, with but little variation, has been done by several arithmeticians in our own country: The Rule for extracting the root depends on the following THEOREM. If line or number be divided into two parts, the cube of the whole line or number, is equal to the cube of the greater part, plus the square of the greater part multiplied by 3 times the less part, plus the square of the less part multiplied by 3 times the larger part, plus the cube of the less part. To illustrate this Theorem, let 27 be divided into two parts, 20 and 7. Then, by the hypothesis, the cube of 27 is equal to the cube of 20, plus the square of 20 multiplied by 3 times 7, plus the square of 7 multiplied by 3 times 20, plus the cube of 7. Hence the following RULE. 1. Separate the given number into periods of three figures each, by putting a point over the unit figure, and every third figure beyond the place of units. 2. Find by the table the greatest cube in the left hand period, and put its root in the quotient. 3. Subtract the cube, thus found, from this period, and to the remainder bring down the next period; call this the dividend. 4. Multiply the square of the quotient by 300, calling it the triple square; multiply also the quotient by 30, calling it the triple quotient; the sum of these call the divisor. 5. Find how many times the divisor is contained in the dividend, and place the result in the quotient. 6. Multiply the triple square by the last quotient figure, and write the product under the dividend ; multiply the square of the last quotient figure by the triple quotient, and place this product under the last; under all, set the cube of the last quotient figure, and call their sum the subtrahend. 7. Subtract the subtrahend from the dividend, and to the remainder bring down the next period for a new dividend, with which proceed as before, and so on, till the whole is completed. Note 1. The same rule must be observed for continuing the operation, and pointing for decimals, as in the square root. NOTE 2. In inquiring how many times the dividend will contain the divisor, we must sometimes make an allowance of two or three units. See National Arithmetic, page 205. 1. What is the cube root of 78402752 ? OPERATION. 4800 120 4920 78402752(428 4x4x300= 64 4x30= 4920) 14402=1st dividend. 1st divisor. 4800 X2= 120x2x2= 8 2x2x2= 10088=1st subtrahend. Ist subtrahend. 9600 480 8 10088 N 158 APPLICATION OF THE CUBE ROOT. [SECT. 49. 530460)4314752=2d dividend. 42x42x300= 529200 4233600 42x30= 1260 80640 2d divisor.= 530460 512 529200 x8=4233600 4314752—2d subtrahend. 1260x8x8= 80640 8x8x8= 512 2d subtrahend.=4314752 2. What is the cube root of 74088 ? Ans. 42. 3. What is the cube root of 185193 ? Ans. 57. 4. What is the cube root of 80621568 ? Ans. 432. 5. What is the cube root of 176558481 ? Ans. 561. 6. What is the cube root of 257259456 ? Ans. 636. 7. What is the cube root of 1860867 ? Ans. 123. 8. What is the cube root of 1879080904 ? Ans. 1234. 9. What is the cube root of 41673648.563 ? Ans. 346.7. 10. What is the cube root of 48392.1516051 ? Ans. 78.51. 11. What is the cube root of 8.144865728 ? Ans. 2.012. 12. What is the cube root of 729? 13. What is the cube root of 498 ? Ans. 37. 14. What is the cube root of 1663 ? 15. What is the cube root of 85%? Ans. 6 Ans. 53. Ans. 44. APPLICATION OF THE CUBE ROOT. Spheres are to each other, as the cubes of their diameter. Cones are to each other, as the cubes of their altitudes or bases. All similar solids are to each other, as the cubes of their homologous sides. 16. If a ball, 4 inches in diameter, weighs 50lbs., what is the weight of a ball 6 inches in diameter ? Ans. 168.7+ lbs. 17. If a sugar loaf, which is 12 inches in height, weighs 16lbs., how many inches may be broken from the base, that the residue may weigh Blbs. ? Ans. 2.5+ in. |