General Rule. In order to extract the cube root of an entire number, divide it into periods of three figures each, commencing on the right, until you arrive at a period containing but one, two, or three figures; extract the root of the greatest cube contained in the first period on the left, and subtract this cube from the first period; bring down to the right of the remainder the first figure of the second period, and divide the number thus formed by three times the square of the figure already found in the root; write the quotient to the right of this figure, and cube the number thus formed; if this cube is greater than the number expressed by the two first periods, diminish the quotient by one or more units, until you obtain a number which can be subtracted from the two first periods. After performing the subtraction, bring down to the right of the remainder the first figure of the third period, then divide the number thus formed by three times the number expressed by the two figures already found; the quotient, if it is not too great, will be such that in writing it to the right of the two figures of the root, and cubing the number thus formed, the result can be subtracted from the number expressed by the three first periods. After this subtraction, bring down to the right of the remainder the first figure of the fourth period, and continue the same operations until you have brought down all of the periods. Remark. In the course of the operation, we may suspect that the quotient of which we have just spoken is much too great, and would wish to diminish it at once by two or more units; but in cubing the root already found, prefixed to this figure, and subtracting this cube from the number expressed by the periods already considered in the given number, we might obtain a very great remainder, which would lead us to suppose that the last figure of the root is too small. But if this is the case, then (157) the remainder must be equal to, or greater than three times the square of the root obtained, plus three times this same root, plus one. In this case the root must be increased by one or more units, of the order of the last figure obtained. To extract the nth root of a whole number. number by N, and the degree When N has not more than n 160. In order to generalize the process for the extraction of roots, we will denote the proposed of the root to be extracted by n. figures, its root has but one, and it is obtained immediately by forming the nth power of each of the whole numbers comprised between 1 and 10, of which the nth power is 10", or the smallest number which contains n+1 figures. When N contains more than n figures, its root has more than one figure, and may then be considered as composed of tens and units. Designating the tens by a, and the units by b, we have (151) n-1 N=(a+b)"=a"+na”—1b+n· an-2b2+, &c.; that is, the proposed number contains the nth power of the tens, plus n times the product of the n-1 power of the tens by the units, plus a series of other parts which it is not necessary to consider. Now as the nth power of the tens cannot give units of an order inferior to unity followed by n ciphers, the n last figures on the right cannot make a part of it. They must then be pointed off, and the root of the greatest nt power contained in the figures on the left should be extracted; this root will be the tens of the required root. If this part on the left should contain more than n figures, the n figures on the right of it must be separated from it, and the root of the greatest n power contained in the part on the left extracted, and so on. After having, in this manner, divided the number N into periods of n figures each, extract the root of the greatest nth power contained in the left hand period; this gives the units of the highest order contained in the total root, or the tens in the root of the number formed by the two first periods on the left. Subtract the nth power of this figure from the left hand period; the remainder, followed by the second period, contains n times the product of the n-1 power of the figure found into the following figure, plus a series of other parts. But it is evident that this first part cannot give units of an order inferior to 10"-1; therefore the n-1 last figures of the second period cannot make a part of it. Hence it is only necessary to bring down to the right of the remainder corresponding to the first period, the first figure of the second period; and if, after having formed n times the n-1 power of the first figure of the root, we divide by this result, the remainder, followed by the first figure of the second period, the quotient will be the second figure of the root, or something greater. To ascertain whether it is too great, we should write it to the right of the first figure, and raise the number thus formed to the nth power; then subtract this result from the two first periods, which will give a new remainder, to the right of which bring down the first figure of the third period; then divide the number thus formed by n times the n-1 power of the two figures of the root already found. This operation is continued until all the periods are brought down. 161. Remark. When the degree of the root to be extracted is a multiple of two or more numbers, as 4, 6, ......, the root can be obtained by extracting the roots of more simple degrees, successively. To explain this, we will remark that (a3) 4a3×a3× a3× a3=a3±3+3+3=a3×4=a12. and that in general (am)"=aTM× aTM × aTM ×a"...=aTM×1(16). Hence the nth power of the mth power of a number, is equal to the mnth power of this number. Reciprocally, the mnth root of a number is equal to the nth root of the mth root of this number, or algebraically .. For, let...√a=a', raising both members to the n er there will result ... 1 m .... pow √a=a'"; (for from the de finition of a root (2), we have (VK)" =K). Again, by raising both members to the mth power, we obtain a=(a'")m=a'mn. Extracting the mnth root of both members, "Wa=a'; but we already have √a=d; hence "a=√a. 3 √2985984=√ √ 2985984= √1728=12; 1679616=√1296=√ √1296=6. N. B. Although the successive roots may be extracted in any order whatever, it is better to first extract the roots of the lowest degree, for then the extraction of the roots of the higher degrees, which is a more complicated operation, is effected upon numbers containing fewer figures than the proposed number. Extraction of Roots by approximation. 162. When it is required to extract the nth root of a number which is not a perfect power, the method of No. 160 will give only the entire part of the root, or the root to within unity. As to the fraction which is to be added in order to complete the root, it cannot be obtained exactly; for the nth power of a frac a tion, cannot be reduced to a whole number, but we can approximate as near as we please to the required root. Let it be required to extract the nth root of the whole num ber a, to within a fraction; that is, so near it, that the error 1 Ρ We will observe that a can be put under the form we denote the root of ap" to within unity, by r, the number n therefore the Va is comprised between these two numbers, that Rule. To extract the root of a whole number to within a 1 fraction, multiply the number by pr; extract the mth root of Ρ the product to within unity, and divide the result by p. 163. Again, suppose it is required to extract the nth root of the Multiply each term of the fraction by -1; it becomes b abr-1 Letr denote the nth root of ab1, to within unity'; a b (r+1)" bn or will be comprised between and bn Therefore, after having made the denominator of the fraction a perfect power of the nth degree, extract the nth root of the numerator, to within unity, and divide the result by the root of the new denominator. When a greater degree of exactness is required than that indicated by, extract the root of ab- to 1 164. Suppose it is required to extract the cube root of 15, to 1 within 12' We have 15 × 123=15×1728=25920. Now the cube root of 25920, to within unity, is 29; hence the required Again, extract the cube root of 47, to within 1 201 We have 47 × 20347 × 8000=376000. Now the cube |