NOTE 1. When a root figure is 0, annex 0 also to the trial divisor, and bring down the next period to complete the new dividend. NOTE 2.—If there is a remainder, after using all the periods in the given example, the operation may be continued at pleasure by annexing successive periods of ciphers as decimals. NOTE 3. -In extracting the root of any number, integral or decimal, place the first point over unit's place; and in extracting the square root, over every second figure from this. If the last period in the decimal periods is not full, annex 0. 2. Find the square root of 46225. 8. Find the square root of 2. 9. Find the square root of 484. 10. Find the square root of 48.4. 11. Find the square root of .064. 12. Find the square root of .00016. Ans. 1.4142+. NOTE. As a fraction is involved by involving both numerator and denominator (Art. 127), the square root of a fraction is the square root of the numerator divided by the square root of the denominator. 13. What is the square root of ? 14. What is the square root of 18? Ans.. 15. What is the square root of? =25. Ans. 2. NOTE. If both terms of the fraction are not perfect squares, and cannot be made so, reduce the fraction to a decimal, and then find the square root of the decimal. A mixed number must be reduced to an improper fraction, or the fractional part to a decimal, before its root can be found. CUBE ROOT OF NUMBERS. 140. The CUBE ROOT of a number is a number which, taken three times as a factor, will produce the given number. 141. The cube of a number consists of three times as many figures as the root, or of one or two less than three times as many. The cube of any number less than 10 must be less than 1000; but any number less than 10 is expressed by one figure, and any number less than 1000 by less than four figures; i. e. the cube of a number consisting of one figure is a number of less than four figures. The cube of any number between 10 and 100 must be between 1000 and 1000000; i. e. must contain more than three figures and less than seven. And in the same way we see that the cube of any number between 100 and 1000 must contain more than six figures and less than ten. Hence, to ascertain the number of figures in the cube root of a given number, Beginning at units, point off the number into periods of three figures each; there will be as many figures in the root as there are periods, and for the incomplete period at the left, if any, one more. 142. To extract the cube root of a number. 1. Find the cube root of 42875. From the preceding explanation, it is evident that the cube root of 42875 is a number of two figures, and that the tens figure of the root is the cube root of the greatest perfect cube in 42; i. e. 27, or 3. Now, if we represent the tens of the root by a and the units by b, a+b will represent the root, and the given number will be = 42875 27000 = 15875. therefore, 3 a2 b + 3 a b2 + b3 But 3 a2b+ 3 a b2 + b3 = (3 a2 + 3 ab+b2) b. If therefore 15875 is divided by 3 a2 + 3 a b+b2 it will give b, the units of the root. But b, and therefore 3 a b + b2, a part of the divisor, is unknown, and we must use 3 a2 divisor. 158752700, or 158 27 = = 2700 as a trial 5, a number that cannot be too small but may be too great, because we have divided by 3 a2 instead of the true divisor, 3 a2 + 3 a b + b2. Then b = 5, and 3 a2 + 3 a b + b2 = 2700 + 450 + 25 = = 3175, the true divisor; 15875, and therefore is the unit's figure of the root, and 35 is the required root. The work will appear as follows: : True divisor, 3 a2 + 3 ab+b2317515875 Hence, to extract the cube root of a number, RULE. Separate the number into periods of three figures each, by placing a dot over units, thousands, &c. Find the greatest cube in the left-hand period, and place its root at the right. Subtract this cube from the left-hand period, and to the remainder annex the next period for a dividend. Square the root figure, annex two ciphers, and multiply this result by three for a TRIAL DIVISOR; divide the dividend by the trial divisor, and place the quotient as the next figure of the root. Multiply this root figure by the part of the root previously obtained, annex one cipher and multiply this result by three; add the last product and the square of the last root figure to the trial divisor, and the SUM will be the TRUE DIVISOR. Multiply the true divisor by the last root figure, subtract the product from the dividend, and to the remainder annex the next period for a dividend. Find a new trial divisor, and proceed as before, until all the periods have been employed. NOTE 1.. -The notes under the rule in square root (Art. 139) apply also to the extraction of the cube root, except that 00 must be annexed to the trial divisor when the root figure is 0, and after placing the first point over units the point must be placed over every third figure from this. NOTE 2. -As the trial divisor may be much less than the true divisor, the quotient is frequently too great, and a less number must be placed in the root. 2. Find the cube root of 18191447. We suppose at first that a represents the hundreds of the root and b the tens: proceeding as in Ex. 1, we have 26 in the root. Then letting a represent the hundreds and tens together, i. e. 26 tens, and b the units, we have a2, the 2d trial divisor, 202800; and therefore b 3; and 3 a = = 3 ab+b2, the 2d true divisor, 205149; and 263 is the required root. NOTE. Though the 1st trial divisor is contained more than 8 times in the dividend, yet the root figure is only 6. 3. Find the cube root of 68716.47. OPERATION. 6 8 7 1 6.4 7 0 (4 0.9 5+ 64 48 0 0.0 0 4716.470 10 8.0 0 .81 4908.81 4417.929 50 18.430029 8.5 41000 6.1 3 50 .0025 5024.5675 25 1.228375 47.31 26 25 |