WRITTEN EXERCISES. 465. 1. What is the cube root of 13824, or what is the edge of a cube whose solid contents are 13824 units? 20%= 13-824(20+4=24 8 000 3 x 202-1200 5 824 3 x 4 x 20 = 240 EXPLANATION.-According to Prin. 2, Art. 463, the orders of units in the cube root of any number may be determined from the number of periods obtained by separating the number into periods containing three Separating the given number thus, there are two periods, or the root is composed of tens and units. 42= 16 1456 5 824 figures each, beginning at units. The tens in the cube root of the number cannot be greater than 2, for the cube of 3 tens is 27000. 2 tens, or 20 cubed, are 8000, which, subtracted from 13824, leave 5824; therefore the root, 20, must be increased by a number such that the additions will exhaust the remainder. The cube (A) already formed from the 13824 cubic units is one whose edge is 20 units. The additions to be made, keeping the figure formed a perfect cube, are 3 equal rectangular solids, B, C, and D; 3 other equal rectangular solids, E, F, and G; and a small cube, H. Inasmuch as the solids, B, C, and D, comprise much the greatest part of the additions, their solid contents will be nearly 5824 cubic units, the contents of the addition. Since the cubical contents of these three equal solids are nearly equal to 5824 units, and the superficial contents of a side of each of these solids are 20 x 20, or 400 square units, if we divide 5824 by 3 times 400, or 1200, since there are 3 equal solids, we shall obtain the thickness of the addition, which is 4 units. Since all the additions have the same thickness, if their superficial contents, or area of each side, are multiplied by 4, the result will be the solid contents of these additions. Besides the larger additions there are three others, E, F, and G, which are each 20 units long and 4 units wide, or whose surfaces have an area of 80 units each, or 240 units altogether; and a small cube whose sides have an area of 16 units. The sum of these areas, 1456, multiplied by 4, the thickness of the additions, gives the solid contents of the additions, which are 5824 units. Therefore the edge of the cube is 24 units in length, or the cube root of 13824 is 24. The tens cannot be greater than 2, for 3 tens raised to the third power is 27000. Cubing 2 tens and subtracting, there is left 5824. This remainder contains 3 times the tens2 x the units + 3 times the tens x the units2, + the units3. 13.824 (24 3txu (20x4) x3= 240 u2=4x4= 16 3t+3tu+u2=1456 Each of these parts contains the units as a factor, hence 5284 is the product of two factors, one of which is the units, and the other 3 times the tens2 + 3 times the tens x the units + the units2. Since 3 times the tens2 is much greater than the rest of the factor, if 5284 is divided by 3 times the tens2, or 1200, the quotient will be the units or other factor. It is found to be 4. The factor completed is therefore 3 x 202 + 3 x 20 x 4 + 42, which is equal to 1200+240 + 16, or 1456. This multiplied by 4 gives the product 5824. Therefore the cube root of the number is 24. STAND. AR.-22 When the number consists of more than two orders of units, the root may be found in the same manner by considering each time the root already found as tens and the next order of the root as units. When the number of figures in the root is more than two the following method materially abridges the process : 4. What is the cube root of 4 to 4 decimal places. EXPLANATION. - Since the root of the number is not a whole number, periods of decimal ciphers are annexed, and the required number of decimal places found. After two figures of the root have been found, the partial divisors may be found as follows: Add together 3 times the product of the tens by the units and the square of the units, then add this sum to the complete divisor, plus the square of the units, and the result, with two ciphers annexed, will be the next partial divisor. Thus, in the example solved, to obtain the partial divisor for the third figure of the root add 150 and 25, and place their sum immediately below the complete divisor. Then add together 175, 475, and 25, and to the sum annex two ciphers. The result will be the next partial divisor. After several decimal places have been found a few more may be found by ordinary division. It will be seen by examining the solution that the numbers added together are equal to 3 ť2 + 6 tu + 3 u2, or 3 (t2 + 2 tu + u2), that is, the sum is 3 times the square of the tens and units already found. RULE.-Separate the number into periods of three figures each, beginning at units. Find the greatest cube in the left-hand period, and write its root for the first figure of the required root. Cube this root, subtract the result from the left-hand period, and annex to the remainder the next period for a dividend. Take three times the square of the root already found, considered as tens, for a partial divisor, and by it divide the dividend. The quotient or the quotient diminished will be the second part of the root. To this partial divisor add three times the product of the first part of the root, considered as tens, by the second part, and also the square of the second part. Their sum will be the complete divisor. Multiply the complete divisor by the second part of the root, and subtract the product from the dividend. Continue thus until all the figures of the root have been found. 1. When there is a remainder, after subtracting the last product annex periods of decimal ciphers, and continue the process. The figures of the root obtained after the ciphers are annexed will be decimals. 2. Decimals are pointed off into periods of three figures each, by beginning at tenths and passing to the right. 3. The cube root of a common fraction is found by extracting the cube root of both numerator and denominator separately, or by reducing it to a decimal and then extracting its root. Extract the cube root of the following: 20. What is the cube root of 2 to four decimal places? 21. What is the cube root of 6 to five decimal places? 22. What is the cube root of ? 5832 ? 157464 ? 19683 34965783 |