PROB. VI. The sum of any two numbers, and their ducts being given, to find each number. pro RULE. From the square of their sum, subtract 4 times their product, and extraet the square root of the remainder, which will be the difference of the two numbers; then half the said difference added to half the sum, 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 sum of the numb. 43 x 43-1849 square of do. Then to the sum of 21,5 [numb. √81-9 diff. of the 41 the diff. Answers. EXTRACTION OF THE CUBE ROOT. A cube is any number multiplied by its square. To extract the cube root, is to find a number, which, being multiplied into its square, shall produce the given number. RULE. 1. Separate the given number into periods of three figures each, by putting a point 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 cube 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 quotient by 300, calling it the divisor. 5. Seck how often the divisor may be had in the divin dend, and place the result in the quotient; then multiply b the divisor by this last quotient figure, placing the product under the dividend. 6. Multiply the former quotient figure, or figures, by the square of the last quotient figure, and that product by 30, 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 subtrahend 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. 1 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 last quotient figure one less; with which find a new subtrahend, by the rule foregoing,) and so on until you can subtract of the subtrahend from the dividend. EXAMPLES. 1. Required the cube root of 18399,744. 18399,744(26,4 Root. Ans. 8 2×2=4×300=1200)10399 first dividend. NOTE-The foregoing example gives a perfect root; and if, when all the periods are exhausted, there happens to be a remainder, you may annex periods of ciphers, and continue the operation as far as you think it necessary. RULE.-1. Find by trial, a cube near to the given number, and call the supposed cube. 2. Then, as twice the supposed cube, 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 exactness. EXAMPLES. 1. Let it be required to extract the cube root of 2. Assume 1,3 as the root of the nearest cube; then-1,8 x 1,3×1,3-2,197 supposed cube. Then, 2,197 2,000 given number. As 6,394 : 6,197 : : 1,3 : 1,2599 roc which is true to the last place of decimas; but might by peating the operation be brought to greater exactres 2. What is the cube root of 584,277056? 3. Required the cube root of 729001101? QUESTIONS, Ans. 900,000 Showing the use of the Cube Root. 1. The statute bushel contains 2150,425 cubic or solid inches. I demand the side of a cubic box, which shall contain that quantity? 3 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 or diameters. 2. If a bullet 3 inches diameter weigh 4 lb. what will a bullet of the same metal weigh, whose diameter is 6 inches ? 3×3×3=27 6×6×6=216. As 27: 4lb.: 216: 32 lb. Ans. 3. If a solid globe of silver, of 3 inches diameter, be worth 150 dollars; what is the value of another globe of silver, whose diameter is six inches? 3x3x3=27 $1200. Ans. 6×6×6=216, As 27: 150 :: 216: The side of a cube being given, to find the side of that cube which shall be double, triple, &c. in quantity to the given cube. RULE.-Cube your given side, and multiply by the given proportion between the given and required cube, and the cube root of the product will be the side sought. EXAMPLES. 4. If a cube of silver, whose side is two inches, be worth 20 dollars; I demand the side of a cube of like silver whose value shall be 8 times as much? 3 2x2x28, and 8x8-6464-4 inches. Ans. 5. There is a cubical vessel, whose side is 4 feet; I de mand the side of another cubical vessel, which shall coptain 4 times as much? 4×4×4=64, and 64×4-256 256-6,349+ft. Ans. A cooper having a cask 40 inches long, and 32 in EVOLUTION, OR EXTRACTION OF ROOTS. 175 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 cask? 3 40×40×40×2=128000 then ✓128000=50,3+length. 82×32×32×2=65536 and 3 65536-40,3+ bung diam. A General Rule for extracting the Roots of all Powers. RULE. 1. Prepare the given number for extraction, by pointing off from the unit's place, as the required root directs. its 2. Find the first figure of the root by trial, and subtract power from the left hand period of the given number. 3. To the remainder bring down the first figure in the next period, and call it the dividend. 4. Involve the root to the next inferior power to that which 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 dividend, and the quotient will be another figure of the root. 6. Involve the whole root to the given power, and subtract it (always) from as many periods of the given number as you have 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 as before, and in like mar.ner proceed till the whole be finished. NOTE. When the number to be subtracted is greater than those periods from which it is to be taken, the last cuotient figure must be taken less, &c. EXAMPLES. 1. Required the cube root of 135796,744 by the above general method. |