either of its sides is equal to its area; of course the square root of its area is equal to the length of either of its sides. 5. When a garden, which is laid out in the form of a square, contains 1,296 square rods, what is the length of each side? that is, what is the square root of 1296? In this example, we know that the root or the length of one side of the garden must be greater than 30, for =900, and less than 40, for 402-1600, which is greater than 1296; therefore we take 30, the less, and, for con venience' sake, write it at the left of 1296 as a kind of divisor likewise at the right of 1296, in the form of a quotient in division; (See Operation 1st. ;) then subtracting the square of 30,900 sq rods, from 1296 square rods, leaves 396 square rods. The pupil will bear in mind that the Fre on the left hand is the form of the garden and contains the same numriber of square rods, viz. 1296. This figure is divided into parts, called A, B, C, and D. It will be perceived that the 900 sq. rds. which we deducted, are found by multiplying the length of A, being 30 rds. by the breadth, being also 30 rods, that is, 302-900. To obtain the square rods in B, C, and D, the remaining parts of the figure, we may multiply the length of each by the breadth of each, thus; 30 x 6=180; 6×6=36; and 30 x 6=180; then 180+36+180=396 square rods; or, add the length of B, that is, 30, to the length of D, which is also 30, making 60; or which is the same thing, we may double 30, making 60; to this add the length of C, 6 rods, and the sum is 66. Now, to obtain the square rods in the whole length of B. C, and D, we multiply their length, 6 rods, by the breadth of each side, thus, 66 x 6=396 square rods, the same as before. We do the same in the operation; that is, we first double 30 in the quotient, and add the 6 rods to the sum, making 66 for a divisor; next, multiply 66, the divisor, by 6 rods, the width, making 396; then tak ng 396 from 396 leaves 0. The pupil will perceive, the only difference between the 1st and 2 operation (which see) is, that in the 2d we neglect writing the ciphers at the right of the numbers, and use only the significant figures. Thus, for 30+6, we write 3 (tens,) and 6 (units,) which, joined together, make 36; for 900, we write 9 (hundreds). This is obvious from the fact, that the 9 retains its place under the 2 (hundreds). Instead of. 60+6, we write 66. Omitting the ciphers in this manner cannot possibly make any difference, and, we see, it does not, fo the result is the same in both. 6. By either of the foregoing operations, then, we find that the length of each side of the garden is 36 rods; or, that the square root of 1296 is 36. 7. PROOF. All the parts of the above figure make as follows, A contains 9 0 0 square rods. D contains 180 square rods. The given sum 1 29 6 square rods. Or by Involution, thus, 36 rods x36 rods=1296 square rods. 8. If then the square of the root, found from the operation, be equal to the given sum, the work is right. 9. Since the square of 99, the greatest factor of two figures, is 9801, which has the same number of figures as both its factors, or only double the number of figures in the root 99, therefore, 10. The square of any root cannot have more figures than double the number of figures in the root. 11. Since the square of 10, the least factor of two places, is 100, which has only one figure less than both its factors, or only one less than double the number of figures in the root, therefore,— 12. The square of any root can never have but one figure less than double the figures of the root. 13. Hence, if we divide any given number into periods of two figures each, the number of periods will equal the number of figures of which the root will consist.. RULE. 14. Point off the given number into periods of two figures each, by puttting a dot over the units, another over the hundreds, and so on; and if there are decimals, point them in the same manner, from the units towards the right hand. 15. Find the greatest square in the last period on the left, write its root on the right, as a quotient, subtract the square from the said period, and to the remainder bring down the next period for a dividend. 16. Double the root (quotient) for a partial divisor, and on its right, place, for the total divisor, such a figure as will express the greatest number of times that the true divisor is contained in the dividend, which figure will be the second in the root, or quotient. 17. Multiply the divisor by the last quotient figure; sultract the product from the dividend; and to the remainder bring down the next period for a new dividend, with which proceed as before, by doubling all the figures in the quotient, or root, &c. Q. How is the operation proved? 8. What is the greatest number of figures which any root can have? 10. What is the least number? 12. What reason is given for each? 9, 11. What is the inference? 13. What is the rule for pointing off the given number? 14. How is the first dividend obtained? 15. What is the direction for finding the second figure in the root? What for finding the next dividend? 17. Repeat the entire rule. 14, 15 16. 16. 17. A. 65536. 22. Extract the square root of 4294967296. 23. When the divisor is too large, increase the dividend by bringmg down the next period of the given sum, then place a cipher in the root, and find a new divisor as before. 2 ) 4 2 0 2 5 ( 2 0 5 Find the sq. root of 42025. 4 4 0 5) 2 0 2 5 2025 Find the sq. root of 651249. A. 205. A. 807. Find the sq. root of 49126081. A. 7009. 24. What is the square root of 6480.25? [See R. 14.] A 80.5. 6 4 1605) 8025 8025 Find the sq. root of 9.3025. A. 3.05. 25. Extract the square root of .0000000001018081. A..00001009. 26. When the last divisor leaves a remainder, the operation may 311 0(3.1 6+ be continued by annexing successively peri Remainder, 1.4 4 Find the sq. rt. of 2. A. 1.41421356237+ 27. When the last period of a decimal consists of only one figure, annex a cipher to complete the period. 28. What is the square root of 11.7? A. 3.4205+ 29. What is the square root of 8.003? 30. What is the square root of .018? A. 2.828+ Q. What is the direction when the divisor is not contained in the dividend 23. What is to be done with the final remainder? 26. decimal period? 27. What with an imperfect 31. When fractions have terms that are perfect powers, [xcvi. 11] extract the roots of the most simple terms. 32. What is the square root of 33. What is the square root of 34. What is the square root of 64 450 ૧ 2048 1234321 121 A. T 1 35. When the terms are either of them imperfect powers, [XCVI.. 12.] reduce them first to a decimal. 11 ? A. .9128+ A. .9198+ A. .83205. A..866+ 36. What is the square root of 7? 37. What is the square root of 38. What is the square root of 39. What is the square root of? 40. What is the square root of? ?? ↑ ? f? Į? ? ? 5 ? !? A. .707+;.816+;.894+ ;.912+;;.755+ ;.745+;.577+. 41. A mixed number may first be reduced to an improper fraction and its roots be expressed again by a mixed number, unless its terms are imperfect powers, in which case the operation must be conducted decimally. 23 1 42. Extract the square root of 4201. 16 A. 2012. A. 30. A. 16. A. 4.1509+ A. 9.35+ A. 15.3196+ A. 9.000862+ 49. If the next divisor or double of the root be written under the final remainder, the fraction will express very nearly the radical remainder, which should be first reduced, either to its lowest terms, or to a decimal, and annexed to the root.* 50. How much is 10946?=104 and 130 rem.: 104×2=208 the next divisor. A. 104130-1045 or 104.625. A. 20798275 415964. A. 736892 1473784. A. 1367. 59. How much are ✓529+✓1764+144+1459264+√6724 Q. What is the rule for extracting the roots of fractions? 31, 35. What for mixed numbers? 41. * Although the remainder is a little too great in the square root, and a little too smal in the cube root, they are nevertheless sufficiently exact for most purposes, and much more convenient than the operation by annexing ciphers. 60. From ✓152399025 take (√4120900++.00060025.) A. 10314.7255 61. What is the square root of 15241578750190521? A. The 9 digits. 62. Find the sum of the roots or numbers involved in all the per fect squares between 1 and 100. A. 44. 63. Find the sum of the squares whose roots are surds, between 1 and 20? A. 160. 64. Suppose that a commandant of an army has 180625 effective men, and would form them into a solid square, how many would there be in each rank and file? A. ✓180625=425. 65. Suppose a town proposes to levy a poll tax of $216.09 so that each man shall pay as many cents as there are men to be taxed; what is each man's tax on his head? A. $1.47. 66. Suppose there are two portions of land each in the form of a square, and that one is 30 miles square, and the other contains 301 square miles; what is the sum of the distances round both squares? A. 143 miles. 67. If the surface of the earth, which is computed to contain 196.000,000 square miles were in the form of a square, what would be the distance round it? A. 56,000 miles. 68. If a tract of land 64 miles long, and 4 miles wide, which cost $1 per acre be exchanged for the same quantity in the form of a square, and subsequently be divided into one hundred equal and square farms, of which should bring at auction $11 per acre; of them $12 per acre, and the rest $10 per acre; what would be the profit in the transaction, and what the sum of the distances round all the squares? A. 200 miles; $164020 profit. PROPORTIONS INVOLVING ROOTS AND POWERS. 69. The product of the square roots of any two numbers, is equal to the square root of their product. 70. Prove 81× √225=√81×225. 71. To find a mean proportional between any two numbers :Extract the square root of their product. 72. For in the proportion 2: 10 :: 10: 50; of which the 10 is a mean proportional between 2 and 50; we have on geometrical principles, 2×50=102. 73. What is the mean proportional between 3 and 12? 4 and 36? 24 and 96? 16 and 64? A. 6; 12; 48; 32. 74. What is the mean proportional between 25 and 289? 25 and 156.25? A. 85; 621 75. What is the mean proportional between 7 and 12? 103 and 41? A. 3: 20.C. Q. To what is the product of the square root of any two numbers equal? 69. How is a mean proportional between any two numbers found? 71 What is the mean proportional between 4 and 9? between 2 and 18? |