down the next period for a dividend, double the root just found for a divisor, and find how often it is contained in this dividend exclusive of the figure on its right hand, annex this quotient to the figures in both the quotient and divisor: multiply the divisor thus completed by the last figure of the quotient, subtract the product as before, and bring down to the remainder the period which comes next in order: repeat the process till every period in succession is disposed of, and the root or an approximation to it will thus be obtained. The divisors tried as above, or the trial divisors, will frequently be taken too large when the dividend consists of only two or three figures, but not so in other cases: and attention to this circumstance will save trouble. Ex. 1. Find the square roots of 1444 and 16129. Proceeding according to the directions given in the Rule, we have or, the square roots of 1444 and 16129 are 38 and 127 respectively and these operations may easily be verified by squaring the numbers 38 and 127: also, the importance of the remark last made will be apparent. Ex. 2. Required the square roots of the mixed decimals 22.09 and 104.7931. = The former of these is a complete square whose root is 4.7; but the latter is not, its approximate root being 10.23 with a remainder .1402: and it will be found upon trial, that (10.23)+.1402 104.7931: also, this approximation might evidently have been carried farther, by affixing to the right hand of the quantity proposed, periods of ciphers which do not affect its value. Ex. 3. Determine the square roots of the fractional From Article (166), we see that the square root of a fraction may be obtained by finding the square roots of its numerator and denominator separately: whence, 144 the square root of 144 12 the square root of = 169 the square root of 169 13 Hence also, since 1278= 31957 the , square root may be found as above: but as the terms are seldom complete squares, it is usual to express the fraction decimally before the rule is applied; and in this instance, we shall have the approximate square root of 1278.28 = 35.753 &c., which might have been extended to more decimal places at pleasure. Ex. 4. Extract the square root of the recurring decimal 1.7. Here 1.7 = and therefore the square root is 16 4 3 1.3: but it generally happens that the corresponding vulgar fraction is not a complete square, and the approximate root must then be found by the ordinary method, though it will not be a recurring decimal. It may here be observed, that the remainder at any stage of the operation must not exceed twice the corresponding quotient or portion of the root: and when a few figures of the root are obtained, their number may nearly be doubled by Division only. Examples for Practice. (1) Find the square roots of 676, 21025, 288369 and 998001. Answers: 26, 145, 537 and 999. (2) Determine the square roots of 2025, 692224, 33016516 and 45859984. Answers: 45, 832, 5746 and 6772. (3) What are the square roots of 5774409, 62805625, 182493081 and 3915380329? Answers: 2403, 7925, 13509 and 62573. (4) Required the square roots of 33.64, 1082.41, 22.8484 and 187.4161. Answers: 5.8, 3.29, 4.78 and 13.69. (5) Find the square roots of .0064, .005329, .00053361 and .00038025. Answers: .08, .073, .0231 and .0195. 3481 (6) Extract the square roots of, 188, and 91. (7) What are the square roots of 435, 108, 345 and 1506111? Answers: 2, 34, 18% and 122. (8) Required the square roots of 321, 411888, .00841 and 756.2816988. 1000 Answers: 53, 6, 2.625, .0029 and 27.5114· 4.41 .64 (9) Determine the square roots of .9, 876.535, 728.6527 and 29.41275 to four places of decimals. Answers: 9486, 29.6063, 26.9935 and 5.4233. (10) What are the square roots of the recurring decimals, .1, .027 and .049382716? Answers: .3, .16 and .2. EXTRACTION OF THE CUBE ROOT. 173. The Investigation of this operation is best conducted by general Symbols, and we shall merely put down here such observations and directions as are necessary and sufficient for performing it. Digits: 1, 2, 3, 4, 5, 6, 7, 8, 9: Cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729: and it is important that these last numbers and the corresponding roots should be committed to memory. 174. Given the number of figures in a number, to find the number of figures in its cube root. Since, the cube root of 1 is 1: the cube root of 1000 is 10: the cube root of 1000000 is 100: &c., it follows that the cube root of a number between 1 and 1000 consists of one figure: that of a number between 1000 and 1000000 of two figures: that of one between 1000000 and 1000000000 of three figures, and so on; so that if a point be placed over every third figure, beginning at the units' place, the number of points thus placed will be that of the digits in the cube root: and it may manifestly be extended to Decimals. Rule for the Extraction of the Cube Root. Point the figures as above directed: then the first figure of the root is the number whose cube is equal to, or next less than, the first period on the left hand: and the remaining figures will be obtained by the following uniform process. To the remainder, if any, bring down the next period, and for a divisor take 300 times the square of the part of the root already found: this gives the next figure of the root: perform the multiplication, to the product add the square of the last figure of the root when multiplied by the rest and by 30, and also the cube of the last, and subtract the sum; to the remainder annex the next period, and proceed in the same way till the root or the requisite approximation to it is obtained. The first and second quotients will frequently be taken too large; the remainder at any step must not exceed three times the square of the root obtained together with three times the root itself, and the number of figures in the root may nearly be doubled by ordinary division. Ex. 1. Extract the cube root of 21952. Here, after pointing the numbers, we have 2 i 9 5 2 (28 = cube root: and this is easily verified, for the cube of 28 = 21952. Ex. 2. Find the cube root of 12812.904. 1 2 8 1 2.9 0 4 (2 3.4 = cube root: 232 × 300 = 158700) 6 4 5 9 0 4 dividend: (1) Determine the cube roots of 1331, 15625, 46656 and 117649. Answers: 11, 25, 36 and 49. (2) Find the cube roots of 2197, 185193, 704969 and 912673. Answers: 13, 57, 89 and 97. |