tinue the operation till all the periods have been used If the final remainder is 0, the root is exact; if not, the root is true to within less than 1. EXAMPLE. 1. Find the cube root of 389017. Here we have two periods. The greatest cube in 389 is 343 (Table) and its cube root is 7. Subtracting 343 from 389, and bringing down 017, pointing off two fig ures, we have 460, which contains 3 x 72 = 147, 3 times. OPERATION. 389 017 147 | 460 17 0. 73 Annexing 3 to 7, we have 73; and cubing, we find the given number, which being subtracted, leaves for a remainder 0. Here we have three periods. The greatest cube in the first is 64, and its root is 4. Subtracting 64 from 111, bringing down the second period and pointing off two figures, we have 479, which contains 3 × 42 = 48, 9 times. Annexing 9 to 4 and cubing 19, we find 117649, which is greater than the first two periods. Cubing 48, we find a result less than the first two periods; hence, 8 is the second figure of the root. Subtracting (48)3 from the first two periods, bringing down the third period and pointing off two figures, we have 13881, which contains 2 (48)2 6912, 2 times. Annexing 2 to 48 and cubing, we find for a result the given number: hence, the cube root required is 482. 3. Find the cube root of 2460375. Ans. 135. Ans. 223. Ans. 739. Ans. 487. 115. The process of extracting the cube root of a number may be generalized. Let n be the index of the root, n being any whole number. We point off the given number into periods of n figures each, beginning at the right the left hand period may contain less than n figures. We next find the greatest nth power in the first period on the left, and write its nth root for the first figure of the required root. We then subtract the nth power of this root from the first period, bring down the next period, and point off n - 1 figures to the right. Dividing the number thus obtained by n times the (n-1)th power of the root found, we get the second figure of the root, or a figure too great. The figure found is tested, and the proof continued as in finding the cube root. EXAMPLE. 1. To find the 5th root of 115856201. 116. When the index of the required root is com posed of two factors, the operation of extracting the root may be simplified. Let N be any number, and assume Raising both members of (1) to the mth power, we Raising both members of (2) to the nth power, we Extracting the mnth root of both members of (3), we Things which are equal to the same thing are equal to each other; hence, placing the first members of (1) and (4) equal, we have, From which, we conclude that the mnth root of any number is equal to the mth root of the nth root of that number. We may, therefore, factor the index and extract that coot of the number indicated by one of the factors, and then that root of the result indicated by the other factor. It will be simplest to begin with the least factor. 117. It has been shown (Art. 100), that a fraction may be raised to any power by raising the numerator to that power for a new numerator, and the denominator to that power for a new denominator. Reversing the principle, we have the rule for finding any root of a fraction. RULE. Extract the required root of the numerator for a ncu numerator, and the same root of the denominator for a new denominator. EXAMPLES. 1. Find the cube root of 2. 2. Find the cube root of. Ans. Ans 3. Find the cube root of 12. Ans. 24 Ans. 2 Ans. 3 Ans. 1 118. If the terms of the fraction are not exact powers of the degree indicated, we may multiply both terms by such a number as will make the denominator an exact power of that degree. Then, extracting the required root of the resulting numerator to within less than 1, and writing the result over the required root of the denominator of the fraction, the result will be the true root, to within less than the fractional unit. 119. The denominator of a decimal fraction is a perfect nth power (n being any whole number), when the number of decimal places is divisible by n, and the number of decimal places in its nth root is the ntb part of the number of decimal places in the given deci |