Solution of Numerical Equations. and the successive values of e, above each line with those h. .1. In the same way may the second and third roots be which has two roots, the first being nearly, 4 and the second Ans. 3.97601 and -4-350577. nearly 4. which has two roots, the first being nearly 1, and the second nearly 2. Ans. 1.0928 and 1.59407. which has three roots, the first being nearly 1, the second nearly 0, and the third nearly-1. Ans. 0.856, 0.3785, —1·2345. 184. Problem. To find any root of a number. Solution. If the required root is the nth root of the number h, this problem is equivalent to solving the equation so that, if the preceding solution is applied to this case, we have u = x", Un x2-1. 185. Corollary. When xa, ma", Mn a"-1, Nn (n-1) an-2 10o e > 10o, (10b e)n > 10n b ; that is, if the root is between 10% and 10%+1 the nth power is between 10b and 10n b+n; or, otherwise, if the left hand significant figure of the root is b places from the decimal point, that of the power must be as many as b times n places from this point, and less than b+1 times n places from it; which, combined with the preceding articles, gives the following rule for finding, the root of a number. Divide the given number into portions or periods beginning with the decimal point, and let each portion or period contain the number of places denoted by the exponent of the power. Find the greatest integral power contained in the left hand period; and the root of this power is the left hand figure of the required root, and is just as many places distant from the decimal point as the corresponding period is removed by periods from this point. Raise the approximate root thus found to the given power and subtract it from the given number, and leave the remainder as a dividend. Raise, again, this approximate root to the power next inferior to the given power, and multiply it by the exponent of the given power for a divisor. The quotient of the dividend by the divisor gives the next figure or figures of the root. Extraction of Roots. The new approximate root, thus found, is to be used in the same way for a new approximation. The number of places to which each division may usually be carried, is so far as to want but one place of doubling the number of places, to which the preceding approximation was found to be accurate. 186. EXAMPLES. 1. Find the fourth root of 5327012345-641. Solution. In the following solution, the columns are the same as the first and second columns in art. 183, except that the top number of the second column is the root which is separated by space into the parts obtained by each successive division, and the number at the top of the first column is divided by spaces into periods. 53 2701 2345 641 2 7* 0.16 37 2701 2345 641 32000000 * This figure must, in the present case, be found by trial, because the first quotient is so inaccurate. Roots of Fractions. Ans. 10.01. Ans. 0.911. Ans. 0.0151. 8. Find the 3d root of 1003-003001. 9. Find the 3d root of 0.756058031. 10. Find the 3d root of 0·000003442951. 11. Find the 5th root of 418227202051. 12. Find the 4th root of 75450765-3376. 13. Find the 5th root of 0·000016850581551. 14. Find the 4th root of 2526-88187761. Ans. 211. Ans. 93.2. Ans. 0.111. Ans. 7.09. Ans. 2.289+. Ans. 3.045+. 187. Corollary. The roots of fractions can be found by reducing them to their lowest terms, and extracting the roots of their numerators and denominators separately. The roots of mixed numbers can be found by reducing them to improper fractions. 189. Corollary. In the case of the square root, we have u = U 2x, m = a2, M = 2 a; and, since the square of a + h is |