2. Required the square root of 184 2. 184 2000(13*57 the root. 1 23 84 36511520 270719500 551 remainder 3. Required the square root of 2 to 12 places. 2(141421356237 + root. 1 24/100 281/400 581/4099 2824/11900 28282 60400 282841 383600 1282841 2828423 10075900 28284238485269 2828426) 1590631 (56237+ ... 1414213 176418 169706 6712 5657 1055 849 206 198 8 4. What is the square root of 152399025 ? 3. What is the square root of '00038754 ? 6. What is the square root of 8. What is the square root of 10? Ans. 12345 Ans. *01809. Ans. 645497. Ans. 2'5298, &c. Ans. 3'162277, kc. TO EXTRACT THE CUBE ROOT. RULE.* 1. Having divided the given number into periods of 3 figures, find the nearest less cube to the first period by the table of powers or trial; set its root in the quotient, and subtract the said cube from the first period; to the remainder bring down the second period, and call this the resolvend. * The reason of pointing the given number, as directed in the rule, is obvious from Cor. 2, to the Lemma, used in demonstrat* ing the square root; and the rest of the operation will be best understood from the following analytical process. Suppose N, the given number, to consist of two periods, and let the figures in the root be denoted by a aad b. Then a+63a3+3a*b+Sab3+63=No= given number, and to find the cube root of Nis the same as to find the cube root of a3+3u36+3ab2+63; the method of doing which is as follows: a3+3a*b+3ab3+b3 (a+6=root. a3 2. To three times the square of the root, just found, add three times the root itself, setting this one place more to the right than the former, and call this sum the divisor. Then divide the resolvend, wanting the last figure, by the divisor, for the next figure of the root, which annex to the former; calling this last figure e, and the part of the root before found call a. 3. Add together these three products, namely, thrice the square of a multiplied by e, thrice a multiplied by the square of e, and the cube of e, setting each of them one place farther toward the right than the former, and call the sum the subtrahend; which must not exceed the resolvend; and if it do, then make the last figure e less, and repeat the operation ibr finding the subtrahend. 4. From the resolvend take the subtrahend, and to the remainder join the next period of the given number for a new resolvend; to which form a new divisor from the whole root now found; and thence another figure of the root, as before, &c. And in the same manner may the root of a quantity, consist ing of any number of periods whatever, be found. 2. What is the cube root of 1092727? 5. What is the cube root of Tt!£ • 6. What is the cube root of? Ans. 103. Ans. 3002. Ans. 05138, &c. Ans. 873 &c. RULE FOR EXTRACTING THE CUBE ROOT BY APPROXIMATION.* 1. Find by trial a cube near to the given number, and call it the supposed cube. * That this rule converges extremely fast may be easily shown thus: Let N given number, a3 supposed cube, and x= correction. Then 2a3+N: 2N+a3 :: a: a+x by the rule, and consequently 2a3 +N×a+x=2N+a3 xa, or 2a3 +a+x3×a+x =2N+a3 xa. Or 2a4+2a3x+a1 +4a3x+6a2x2 +4ax3 +x=2aN+a“, and by transposing the terms, and dividing by 2a 304 2a' N=a3 +3a2x+3ax2+x3+x3+, which by neglecting the VOL. I. Τ 2. Then twice the supposed cube added to the given number is to twice the given number added to the supposed cube, as the root of the supposed cube is to the root required nearly. Or as the first sum is to the difference of the given and supposed cube, so is the supposed root to the difference of the roots nearly. 3. By taking the cube of the root thus found for the supposed cube, and repeating the operation, the root will be had to a still greater degree ol exactness. EXAMPLES. 1. It is required to find the cube root of 98003449. Let 125000000= supposed cube, whose root is 500; Then 125000000 98003449 348003449)160503449000(461=corrected root, or 1392013796 2130206940 2088020694 421862460 348003449 73859011 2. Required the cube root cf 21035'8. Here we soon find that the root lies between 20 and 30, 24 2a' terms 3 + as being very small, becomes. N=a3 +3a2x+ 3 Saxx3 the known cube of a+x. Q. E. I. |