Pron. VI. The sum of any two numbers, and their proJucts being given, to find each number. Rule.-From the square of their sum, subtract 4 times their product, and extract the square root of the remainder, which will be the difference of the two numbers; then half the said difference added to half the suin, gives the greater of the two numbers, and the said half difference subtracted from the half sum, gives the lesser number. EXAMPLES. The sum of two numbers is 43, and their product is 442; what are those two numbers ? The suin of the numb. 43 X 43=1849 square of do. The product of do. 442 x 4=1768 4 times the pro. Then to the sum of 21,5 (numb. .tand 4,5 V81-29 diff. of the EXTRACTION OF THE CUBE ROOT. A cube is any number multiplied by its square. To extract the cube rout, is to find a number, which, being multiplied into its square, shall produce the given num, ber. RULE. 1. Separate the given number into periods of three figures each, by putting a posi'nt over the unit figure, and every third figure from the place of'units to the left, and if there be decimals, to the right. 2. Find the greatest cube in the left hand period, and place its root in the quotient. 3. Subtract the cuhe thus found, from the said period, and to the remainder bring down the next period, calling this the dividend. 4. Multiply the square of the quotien, by 300, calling it the divisor. 5. Seek how often the divisor may be had in the dins dend, and place the result in the quotient; then multiple the divisor by this last quotient figure, placing the produce under the dividend. 6, Multiply the former quotient figure, or figures, by the square of the last quotient figure, and that product by 311, and place the product under the last; then under these two products place the cube of the last quotient figure, and add them together, calling their sum the subtrahend. 7. Subtract the subirahend from the dividend, and to the remainder bring down the next period for a new dividend; with which proceed in the same manner, till the whole be finished, Note.--If the subtrahend (found by the foregoing rule happens to be greater than the dividend, and con 'equently cannot be subtracted therefrom, you must make the lası quotient figure.one less; with which find a new subtralied (by the rule foregoing,) and so on until you can subtrac' the subtrahend from the dividend. EXAMPLES. 1. Required the cube root of 18399,744. 18399,744(26,4 Root. Ans. 2x2=4x300=1200)10399 first dividend 7200 6x636 * 2.672x30=2160 6x6x6= 216 9576 Įst subtrahenú. 36X26=676 X 300202800)823744 ?d dividend. 811200 4*4=16 x 26=416 x 30— 12480 4X4X4= 64 823744 20 subtrahan Note.--The foregoing example gives a perfect root ; And if, when all the periods are exhausted, there happens co he a remainder, you may annex periods of ciphers, and cortinue the operation as far as you think it necessary. Answers 2. What is the cube root of 205379 ? 59 3. Of 614125? 85 41421736 ? 316 5. Of 146363,183 ? 6. Of 29,508381 ? 3,09+ 7. Of 80,763 ? 8. Of ,162771336 ? ,546 9. Of ,000684134? ,088+ 10. Of 122615327232 ? 4968 4. Of 52,7 4,32+ Rule.-1. Find by trial, a cube near to the given number, and call it che supposed cube. 2. Then, as twice the supposed cubo, added to the given number, is to twice the given number added to the supposed cube, so is the root of the supposed cube, to the true root, or an approximation to it. 3. By taking the cube of the root thus found, for the supposed cube. and repeating the operation, the root will be had to a greater degree of uxactness. EXAMPLES, 1. Let it be required to extract the cube root of 2. Assume 1,3 as the root of the nearest cube ; then-1,3% 1,3 x 1,3=2,197=supposed cube. Then, 2,197 2,000 given number. 2 2 4,394 4,000 2,000 2,197 As 6,394 6,197 : 1,3 : 1,2592 roo', which is true to the last place of decinan: but might by r. peating the operation he brought t meet 1 (rictulas:. 2. What is the cube root of 584,9 * : : 3 3. Required the cube root of 729001101 ? Ans, 900,0004 QUESTIONS, Showing the use of the Cube Root. 1. The statute bushel contains 2150,425 cubic or solu ánches. I demand the side of a cubic box, which shall con: tain that quantity ? V 2150,425=12,907 inch, Ans. =. Note.—The solid contents of similar figures are in proportion to each other, as the cubes of their similar sides oi diarneters. 2. If a bullet 3 inches diameter weigh 4 lb. what will : bullet of the same metal weigh, whose diameter is 6 in ches? 3x3x3-27 6x6x6=216. As 27 : 4 lb. : : 216 32 lb. Ans. 3. If a solid globe of silver, of 3 inches diameter, in worth 150 dollars; whạt is the value of another globe o! silver, whose diameter is six inches? 3x3x3=27 6x6x6=216, As 27 : 150 :: 216 $1200. Ans. The side of a cube being given, to find the side of tha cube which shall be double, triple, &c. in quantity to thu given cube. RULE.-Cube your given side, and multiply by the given propor tion between the given and required cube, and the cube root of th product will be the side sought. EXAMPLES. 3 4. If a cube of silver, whose side is two inches, be wort! 20 dollars; I demand the side of a cube of like silver whose value shall be 8 times as much ? 2x2x28, and 8x8=64764=4 inches. Ans. 5. There is a cubical vessel, whose side is 4 feet; I de mand the side of another cubical vessel, which shall con gain 4 times as much ? 4x4x4=64, and 64 x4=256 256–6,349+ft. Ans. D. A cuoper having a cask 40 inches long, and 32 in ches at the bung diameter, is ordered to make another cask of the same shape, but to hold just twice as much; wha will be the bung diameter and length of the new cusk ? 40 x 40 x 40x2=128000 then 128000=50,3+length. 32 x 32 x 32 x2=65536 and P65536=40,3+ bung diam. A General Rule for extracting the Roots of all Powers. RULE. 1. Prepare the given number for extraction, by pointing ' from the unit's place, as the required root directs. 2. Find the first figure of the root by trial, and subtract tis power from the left hand period of the given number. 3. To the remainder bring down the first figure in the Wext period, and call it the dividend. 4. luvolve the root to the next inferior power to that hich is given, and multiply it by the number denoting the given power, for a divisor. 5. Find how many times the divisor may be had in the Hiyidend, and the quotient will be another figure of the root. 6. Inyolye the whole root to the given power, and subtrace it (always) from as many periods of the given number as you haye found figures in the root. 7. Bring down the first figure of the next period to the remainder for a new dividend, to which find a new divisor us before, and in like mariner proceed till the whole be finished. Nute. When the number to be subtracted is greater than those periods froin which it is to be taken, the last "Hotient figure inust be taken less, &c. EXAMPLES, 1. Required the cube root of 135796,744 by the above general inethod. |