of It is evident, as evolution is the reverse of involution (Art. 517), that from the process now given of obtaining a square may be deduced a method of extracting its root. Since the square (a+b) is a + 2ab + b?, the square root of a + 2ab +62 must be a + b. Now it will be observed that a, the first term of the root, is the square root of a’, the first term of the square ; and if a’ be subtracted, there will remain 2 ab + 1, from which b, the second term of the root, is to be obtained. But 2 ab + b? is the same as (2 a +b) X b, therefore the remainder equals (2 a + b) X b. But as b, the units, is always much less than 2 a, twice the tens, we consider that 2 a X b is about equal to the whole remainder, and taking 2 a (which we know) as the trial divisor, we obtain b, the units. But as the true divisor is 2 a + b, we add the units to twice the tens and multiply the sum by the units, which gives a product equal to the whole remainder, or 2 ab + b?. 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. 525. To extract the square or second root of numbers. OPERATION. Ex. 1. What is the square root of 1296 ? Ans. 36. Beginning at the right, we separate the 1 29 6 36 number into periods of two figures each, by placing a point (1) over the right-hand figure 9 of each period. Since the number of periods 6 6 3 9 6 is two, the root will consist of two figures, 3 9 6 tens and units. Then 1296 the square of the tens plus twice the product of the tens 0 into the units, plus the square of the units. The square of tens is hundreds, and must therefore be found in the hundreds of the number. The greatest number of tens whose square does not exceed 12 hundreds is 3, which we write as the tens figure of the root. We subtract the 9 hundreds, the square of the 3 tens, from the 12 hundreds, and there remain 3 hundreds; after which we write the figures of the next period, and the remainder is 396 twice the product of the tens into the units plus the square of the units. We have then next to find a number which, added to twice the 3 tens of the root, and multiplied into their sum, shall equal 396. By dividing this remainder OPERATION. by twice the three tens of the root, we may obtain the units, a number somewhat too large. But though it may be too large, it cannot be too small, since the remainder 396 contains twice the product of the tens into the units, and also the square of the units. We therefore make twice the three tens of the root = 6 tens, a trial divisor, with which we divide the 39 tens, exclusive of the 6 units, which cannot form any part of the product of the tens by the units. The quotient figure obtained, 6, must be the units figure of the root, or a number somewhat larger. To determine whether it expresses the real number of the units in the root, we annex it to the 6 tens, and multiply the number 66, thus formed, by it. The product is 396, which being subtracted, there is no remainder. Therefore 1296 is a perfect square, and 36 the root sought. 2. What is the square root of 278784? Ans. 528. Since there are three pe2 1 818 4 5 28 riods, the root will contain 2 5 three figures; the first two may be considered as tens 102 2 87 and units of TENS. As the 204 square of tens cannot give 10 4 8 8 3 8 4 less than hundreds, we must find that square in the two 83 84 left-hand periods; and as we 0 have tens and units of tens, their 528 x 528 278784 the tens plus twice the tens into the units, plus the square of the units 2787 (nearly). We proceed then with the first two periods exactly the same as when the root consists of but two figures, and thus take from the given number the square of the 52 tens, which leaves a remainder of 8384. We now consider the given number 278784, as the square of a number consisting of 52 tens and a certain number of units, which square will of course equal the square of the tens plus twice the tens into the units, plus the square of the units. But the square of the tens, or (52)? has already been taken from the given number, leaving a remainder, 8384, which must equal twice the tens into the units plus the square of the units. From this we readily obtain the units, just as when we had but two figures in the root. RULE. — Separate the given number into as many periods as possible of two figures each, by placing a point over the place of units, another over the place of hundreds, and so on. Find the greatest square in the left-hand period ; write the root of it at the right of the given number after the manner of a quotient in division, and subtract the second power from the left-hand period. Bring down the next period to the right of the remainder for a dividend, and double the root already found for a trial divisor. Find how often this divisor is contained in the dividend, exclusive of the righthand figure, and write the quotient as the next figure of the root. PROOF square = the square of 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. EXAMPLES. Ans. At 3. What is the square root of iltig? Vinttg = = 41 4. What is the square root of go? Ans. .1936+ V240 80 Or =.0375; V.0375 .1936+ 5. What is the square root of 3444736 ? Ans. 1856. 6. What is the square root of 998001 ? Ans. 999. 7. What is the square root of 340g ? 8. Extract the square root of 234.09 ? Ans. 15.3. 9. What is the square root of 424?' Ans. 64 10. What is the square root of .000729 ? Ans. 027. 11. What is the square root of 17.3056? Ans. 4.16. 12. What is the square root of 52 16 Ans. 74 13. What is the square root of 9516? Ans. 9 14. What is the square root of 3637_1? Ans. 1915. 15. How much is v1.96 ? Ans. 1.4. 16. How much is 65611? Ans. 81. 17. How much is „93 ? Ans. 27. 18. How much is gi? Ans. 64. 19. How much is one of the two equal factors of 9645192360241 ? 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 requireil, 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 ? Ans. 28.01785. CONTRACTED METHOD. COMMON METHOD. 7 8 5 2 8.0 1785+ 4 3 84 5 601 7 8 5 2 8.0 1 785+ 4 48. 385 384 5 6 0110000 5 601 39 2 1 9 9 5 6 0 2 7 14 39 900 392 1 8 9 448 2 784 280 178 2 5 8137 7 5 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 6; 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 decimals. Ans. 1.77482393+. 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. times as many. 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 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 |