CUBE ROOT OF NUMBERS. 210. The method of Art. 209 may be used to extract the cube roots of arithmetical numbers. The cube root of 1000 is 10; of 1,000,000 is 100; etc. Hence, the cube root of a number less than 1000 is less than 10; the cube root of a number between 1,000,000 and 1000 is between 100 and 10; and so on. That is, the integral part of the cube root of a number of one, two, or three figures, contains one figure; of a number of four, five, or six figures, contains two figures; and so on. Hence, If a point be placed over every third figure in any integral number, beginning with the units' place, the number of points shows the number of figures in the integral part of its cube root. 211. Let it be required to find the cube root of 157464. 157464 α3 = 125000 50+4 = a+b 3a2 = 7500 32464 3 ab = 600 b2= 16 8116 32464 Pointing the number according to the rule of Art. 210, we see that there are two figures in the_integral part of the cube root. Let a denote the value of the number in the tens' place in the root, and b the number in the units' place. Then a must be the greatest multiple of 10 whose cube is less than 157464; this we find to be 50. Subtracting a3, that is, the cube of 50 or 125000, from the given number, the remainder is 32464. Dividing this remainder by 3a2, that is, 3 times the square of 50 or 7500, we obtain 4 as the value of b. Adding to the trial-divisor 3 ab, that is, 3 times the product of 50 and 4, or 600, and b2, or 16, we have the complete divisor 8116. Multiplying this by 4, and subtracting the product, 32464, there is no remainder. Hence, 50+ 4 or 54 is the required cube root. The ciphers being omitted for the sake of brevity, the work will stand as follows: 157464 | 54 7500 32464 600 16 8116 32464 From the above process, we derive the following rule: Separate the number into periods by pointing every third figure, beginning with the units' place. Find the greatest cube in the left-hand period, and write its cube root as the first figure of the root; subtract its cube from the number, and to the result bring down the next period. Divide this remainder by three times the square of the root already found, with two ciphers annexed, and write the quotient as the next figure of the root. Add to the trial-divisor three times the product of the last root-figure and the part of the root previously found, with one cipher annexed, and the square of the last root-figure. Multiply the complete divisor by the figure of the root last obtained, and subtract the product from the remainder.. If other periods remain, proceed as before, taking three times the square of the root already found for the next trial-divisor. The notes to Art. 206 apply with equal force to examples in cube root, except that in Note 3 two ciphers should be annexed to the trial-divisor. 212. In the illustration of Art. 208, if there had been more terms in the given quantity, the next trial-divisor would have been three times the square of a+b; that is, 3a2+6ab+3b. We observe that this is obtained from the preceding complete divisor, 3a2+3ab+b2, by adding to it its second term, 3 ab, and twice its third term, 20o. We may then use the following rule for forming the successive trialdivisors in the cube root of numbers: To the preceding complete divisor, add its second term anā twice its third term; and annex two ciphers to the result. Since the second root-figure is 0, we annex two ciphers to the trialdivisor 1200, and bring down to the remainder the next period, 865. The second trial-divisor is formed by the rule of Art. 212. The preceding complete divisor is 120601; adding its second term, 600, and twice its third term, 2, we have 121203; annexing two ciphers to this, we obtain the result 12120300. Extract the cube roots of the following: 2. 29791. 3. 97.336. 4. .681472. 5. 1860867. 6. 1.481544. 7..000941192. 8. 8.242408. 9. 51478848. 10. 10077.696. 12. 116.930169. 13..031855013. 14. .724150792. 15. 1039509.197. 16. .000152273304. Extract the cube roots of the following to four figures: 214. When the index of the required root is the product of two or more numbers, we may obtain the result by successive extractions of the simpler roots. For, by Art. 198, (mn/a)mn = α. Taking the nth root of both members, (mn/a)m = a. Taking the mth root of both members of (1), That is, mn m /a="/(/α). (1) The mnth root of a quantity is equal to the mth root of the nth root of that quantity. For example, the fourth root is the square root of the square root; the sixth root is the cube root of the square root; etc. EXAMPLES. Find the fourth roots of the following: 1. 16 x2 - 96 x2y + 216 x2y2 - 216 ху3 + 81 у*. 2. x-4x2+10x - 16x + 19 x2 - 16 x2 + 10 x2 - 4x + 1. 3. x-8x2+16x+16x5-56 x2-32x2+64x2+64x+16. Find the sixth roots of the following: 4. α12 - 6a10 + 15 a8 – 20 a2 + 15 a* - 6 a2 + 1. 5. 64x+192 x + 240x2 + 160x2 + 60x2 + 12x + 1. XIX. THE THEORY OF EXPONENTS. 215. In the preceding chapters we have considered an exponent only as a positive whole number. It is, however, found convenient to employ fractional and negative exponents; and we proceed to define them, and to prove the rules for their use. 216. In Art. 13 we defined a positive integral exponent as indicating how many times a quantity was taken as a factor; thus, am signifies a xaxax・・・ to m factors. We have also found the following rules to hold when m and n are positive integers : I. am xa" = am+n. II. (am)" = amn. (Art. 79.) (Art. 193.) It 217. The definition of Art. 13 has no meaning unless the exponent is a positive integer, and we must therefore adopt new definitions for fractional and negative exponents. is convenient to have all forms of exponents subject to the same laws in regard to multiplication, division, etc., and we shall therefore assume Rule I. to hold for all values of m and n, and find what meanings must be attached to fractional and negative exponents in consequence. 218. Required the meaning of a. Since Rule I. is to hold universally, we must have 5 a × a × a3 = a3++ = a. That is, as is such a quantity that when raised to the third Hence (Art. 198), a must be the power the result is a. cube root of a; or, a a = Vď. |