Reduce the fraction to its lowest terms for this and all other roots; then 1. Extract the root of the numerator for the new numerator, and the root of the denominator, for a new denominator. 2. If the fraction be a surd, reduce it to a decimal, and extract its root. EXAMPLES. 162 1024 1. What is the square root of 98 ? Answers 154/ 9128+ 7. What is the square root of 2? ,7745+ 8. Required the square root of 361 ? 6,0207+ 6. What is the square root of APPLICATION AND USE OF THE SQUARE ROOT. PROBLEM I. A certain General has an army of 5184 men; how many must he place in rank and file, to form them into a square? RULE. Extract the square root of the given number. 5184-72 Ans. PROB. II. A certain square pavement contains 20736 square stones, all of the same size; I demand how many are contained in one of its sides ? ✓/20736=144 Ans. PROB. III. To find a mean proportional between two numbers. RULE. Multiply the given numbers together, and extract the square root of the product. EXAMPLES. What is the mean proportional between 18 and 72? 1296-56 Ans." 72X18-1296, and PROB. IV. To form any body of soldiers so that they may be double, triple, &c. as many in rank as in file. RULE. Extract the square root of 1-2, 1-3, &c. of the given number of men, and that will be the number of men in file, which double, triple, &c. and the product will be the number in rank. EXAMPLES. Let 13122 men be so formed, as that the number in rank may be double the number in file. = 13122÷2-6561, and 162 in rank. 6561-81 in file, and 81x2 PROB. V. Admit 10 hds. of water are discharged through a leaden pipe of 24 inches in diameter, in a cer tain time; I demand what the diameter of another pipe must be, to discharge four times as much water in the same time. RULE. Square the given diameter, and multiply said square by the given proportion, and the square root of the pro duct is the answer. 21=245, and 2,5×2,5=6,25 square. 4 given proportion. ✔25,00=5 inch. diam. Ans. के PROB. VI. The sum of any two numbers, and their products 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 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. 43×43=1849 square of do. The product of do. 442x 41768 4 times the pro. Then to the sum of 21,5 [numb. 81-9 diff. of the +and 4,5 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. Seek how often the divisor may be had in the div dend, and place the result in the quotient; then multiply 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. NOTE. If the subtrahend (found by the foregoing rule) happens to be greater than the dividend, and consequently 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 the subtrahend from the dividend. EXAMPLES. 1. Required the cube root of 18399,744. 8399,744(26,4 Root. Ans. 8 2×2=4×300-1200) 10399 first dividend. 7200 6×6=36×2=72×30=2160 6×6×6= 216 9576 1st subtrahend. 811200 26×26-676×300-203800)823744 2d dividend. 4x4-16x26=416×30= 12485 4X4X4= 64 823744 2d subtrahend. 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 cyphers, and continue the operation as far as you think it necessary. Answers. 2. What is the cube root of 205579? 59 1. Find by trial, a cube near to the given number, and call it 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 extmen EXAMPLES. Let it be required to extract the cube root of 2. Assume 1,3 as the root of the nearest cube; then1,3×1,3×1,3 2,197 supposed cube. Then, 2,197 2,000 given number. As 6,394 : 6,197 : : 1,3 : 1,2599 root, which is true to the last place of decimals; but might by repeating the operation, be brought to a greater exactness. 2. What is the cabe root of 584,277056 P Ans. 8,36. |