Annex the last root figure to the trial divisor for the true divisor, which multiply by the last root figure and subtract the product from the dividend. To the remainder bring down the next period for a new dividend. Double the root already found for a new trial divisor, and continue the operation as before, till all the periods have been brought down. NOTE 1. When the product of any trial divisor exceeds its corresponding dividend, the last root figure must be made less. If a dividend does not contain its corresponding divisor, a cipher must be placed in the root, and also at the right of the divisor; then, after bringing down the next period, this last divisor must be used as the divisor of the new dividend. NOTE 2. When there is a remainder after extracting the root of a number, periods of ciphers may be annexed, and the figures of the root thus obtained will be decimals. NOTE 3. If the given number is a decimal, or a whole number and a decimal, the root is extracted in the same manner as in whole numbers, except, in pointing off the decimals, either alone or in connection with the whole number, we place a point over every second figure toward the right, from the separatrix, filling the last period, if incomplete, with a cipher. The number of decimal places in the root will always equal the number of periods of decimals in the power. NOTE 4. - If the given number is a common fraction, reduce it to its simplest form, if it is not so already, and extract the root of both terms, if they are perfect powers; otherwise, either find their product, extract its root, and divide the result by the denominator, or reduce the fraction to a decimal, and extract the root of the decimal. NOTE 5. When the given number is a mixed number, it may be changed to the form of a common fraction, or the fractional part may be reduced to a decimal, before attempting to extract the root. Or g .0375; .0375 = = .1936+. 5. What is the square root of 3444736? Ans. 1856. Ans. 999. Ans. 15.3. Ans. 64. 19. How much is one of the two equal 9645192360241? Ans. 64. factors of Ans. 3105671. 526. When the square root is to be extracted to many places of figures, the work may be contracted thus: — Having found in the usual way one more than half of the root figures required, the rest may be found by dividing the last remainder, with a single figure annexed instead of two, by the last divisor, and proceeding as in contracted division of decimals. (Art. 276.) EXAMPLES. 1. What is the square root of 785 to five places of decimals? The nature and extent of the contraction will be seen by comparing the contracted method with the common method. 2. Extract the square root of 63 to four places of decimals. Ans. 2.5298+. 3. Required the square root of 2 to five places of decimals. Ans. 1.41421+. 4. Required the square root of 3.15 to eight places of deciAns. 1.77482393+. mals. 5. Required the square root of 373 to seven places of decimals. Ans. 19.3132079+. 6. Extract the square root of 8.93 to eight places of decimals. Ans. 2.98831055+. EXTRACTION OF THE CUBE ROOT. 527. The extraction of the cube root of a number is the process of finding one of its three equal factors; or, of finding a factor which, being multiplied into itself twice, will produce the given number. 528. The common method of extracting the cube root depends upon the following principles: = = 1. The cube of any number has, at most, only three times as many figures as its root, and, at least, only two less than three times as many. For the cube of a number of a single figure consists of, at most, three figures, and, at least, two less than that number, as 13 1, and 93 = 729; the cube of a number of two figures consists of, at most, six figures, and, at least, two figures less than that number, as 103 = 1000, and 993 970299; and so on. Therefore, when a cube number consists of one, two, or three figures, its root will consist of one figure; when of four, five, or six figures, its root will consist of two figures, and so on; and if a number be separated into as many periods as possible of three figures, each commencing at the right, to these periods respectively will correspond the units, tens, hundreds, &c. of the cube root of that number. 2. The cube of a number consisting of TENS and UNITS is equal to the cube of the tens, plus three times the square of the tens into the units, plus three times the tens into the square of the units, plus the cube of the units. Thus, if the tens of a number be denoted by a, and the units by b, the cube of the number will be denoted by (a+b)3 = a3 + 3 a2 b + 3 a b2 + b3. Then, by this formula, if a = 3, and b equal 6, we have 3 tens +6 units 30+6 = 36, and 363 (30+6)3 = 303+3 (302 × 6)+3(30 × 62) +63 = 46656. Or, analytically, (a+b)2 = 303 +2 x 30 x 6 +62 a+b =30+6 (a+b)2xa=303+2 × 302 × 6+30 × 62 (a+b)2xb= 302 × 6+2×30×62+63= 7776 (a+b)3 = 302+3 (302 × 6)+3 (30×62)+63-46656 But It is evident, as evolution is the reverse of involution, that from this process of obtaining a cube may be deduced a method of extracting the cube root. Since the cube of a + b is a3 + 3 a2 b + 3 a b2 + b3, the cube root of a3 + 3 a2 b + 3 a b2+b3 must be a+b. Now a, the first term of the cube root, is the cube root of a3, the first term of the cube; and if a3 be subtracted, there will remain 3 a2 b + 3 a b2 + b3, from which b, the second term of the root, is to be obtained. 3 a2 b + 3 a b2 +63 is the same as (3 a2 + 3 a b + b2) × b ; therefore the remainder equals (3 a2 + 3 a b + b2) × b. But as 3 ab, three times the tens into the units, plus b2, the square of the units, is generally much less than 3 a2, three times the square of the tens, we consider that 3 a b is about equal to the whole remainder, and taking 3 a2 (which we know) as the trial divisor, we obtain b, the units. But as the true divisor is 3 a2 + 3 a b + b2 we add three times the tens by the units plus the square of the units, and multiply the sum by the units, which gives a product equal the whole remainder, or 3a2b3ab2 + b3. Since every number of more than one figure may be considered as composed of tens and units, we may have tens and units of units, tens and units of tens, tens and units of hundreds, &c. Hence, the principle just explained applies equally whether the root contains two or more than two figures. 529. To extract the cube or third root of numbers. = riods, by placing third a point over the tens and units. Then 46656 the cube of tens, plus three times the square of the tens into the units, plus three times the tens into the square of the units, plus the cube of the units. The cube of tens is thousands, and must therefore be found in the thousands of the number. The greatest number of tens whose cube does not exceed 46 thousands is 3, which we write as the tens figure of the root. We then subtract the 27 thousands, the cube of the 3 tens, from the 46 thousands, and there remain 19 thousands; and, annexing the next period, we have as the entire remainder, 19656, equal three times the square of the tens into the units, plus three times the tens into the square of the units, plus the cube of the units, or the product of three times the square of the tens, plus three times the tens into the units, plus the square of the units, multiplied by the units. By dividing this remainder by three times the square of the tens of the root, we obtain the units, or a number somewhat too large. Although it may be too large, it cannot be too small, since the remainder 19656 contains not only three times the square of the tens into the units, but three times the tens into the square of the units, plus the cube of the units. We therefore make three times the square of the tens of the root, = 27 hundreds, a trial divisor, with which we divide the 196 hundreds of the remainder, disregarding the 56 units, since they cannot form any part of the product of the square of the tens by the units. The quotient figure obtained, 7, must be the units figure of the root, or a number somewhat larger. But on undertaking to complete the divisor on the supposition that 7 is the true units figure of the root, we find a divisor too large for the remainder. We therefore take 6, a number one less, and to determine whether it expresses the real number of units in the root, we add to the 27 hundreds of the trial divisor three times the 3 tens of the root into the 6 units, plus the square of the 6 units; and multiplying the true divisor, 3276, thus formed, by the units, and subtracting the product, 19656, from the remainder, there is nothing left. Hence, 46656 is a perfect cube, and 36 its cube root. |