168 Evolution, OR Extnaction of Roots. given number, (after the manner of a quotient in division,) for the first figure of the root, and the square number under the period, and subtract it therefrom, and to the re mainder bring down the next period, for a dividend. 3. Place the double of the root, already found, on the left hand of the dividend, for a divisor. 4. Place such a figure at the right hand of the divisor, and also the same figure in the root, as when multiplied into the whole (increased divisor) the product snail be equal to, or the next less than the dividend, and it will be the second figure in the root. 5. Subtract the product from the dividend, and to the remainder join the next period for a new dividend. 6. Double the figures already found in the root, for a new divisor, and from these find the next figure in the root as last directed, and continue the operation in the same manner till you have brought down all the periods. Or, to facilitate the foregoing Rule, when you have brought down a period, and formed a dividend in order to find a new figure in the root, you may divide said dividend (omitting the right hand figure thereof) by double the roof already found, and the quotient will commonly be the figures sought, or being made less one or two, will generally give the next figure in the quotient. Ex-MP-Es. 1. Required the square root of 141225,64. 1412:25,64(375,8 the root exactly without a remainder; 9 but when the periods belonging to any - given number are exhausted, and still 67)512 leave a remainder, the operation may 169 be continued at pleasure, by annexing - periods of ciphers, &c. 745)4325 3725 750S)00064 60064 0 remains. TO Evolution, on Extraction of Roots. 109. What is the square root of 12961 answers. 36 Of - 566441 23,8 Of - 5-1900257 2.345 Ot - 3537.2961-1 - titol Of - 184,21 13,57+ Of - 97.12,6938097 98,553 Of - 0,45369? ,673+ Ot - 00:2916? ,05: Of - 45.1 6,708+ EXTRACT THE SQUARE ROOT OF WUL- RULE. Reduce the fraction to its lowest terms for this and all other roots; then 1. Extract the root of the numerator for a new numera or, and the root of the denominator, for a new denominator. 2. If the fraction be a surd, reduce it to a decimal, and 1. What is the square root of to Answers. ; 2. What is the square root of or ? o 3. What is the square root of +441 4. What is the square root of 2011 4+ 5. What is the square root of 248, 1 15: SURDS. 6. What is the square root of of 1 91287. What is the square root of #1 ,7745+ 8. Required the square root of 364 6,0207+ APPLICATION AND USE OF THE SQUARE, ROOT. Paontext L-A certain general has an army of 5184 men; how many unust he place in rank and file, to form them into a square? - 10 Evolution, on Extraction of Roots. Rult.—Extract the square root of the given number. vol.84=72 Ans. Prop. 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? v20.736–144 Ans, Prop. III. To find a mean proportional between two numbers. Rule.—Multiply the given numbers together and extrao the square root of the product. Ex-MPLEs. What is the mean proportion all between 18 and 72? 7:22, 18–1296, and v. 1296–36 Ans. Prou. 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. Ex-MPLEs. Let 13122 men be so formed, as that the number in rank may be double the number in file. 13122-2–6561, and v5561=81 in file, and 81 ×2 -162 in rank. Pros. W. Admit 10 bhds. of water are discharged through a leaden pipe of 24 inches in diameter, in a certain time; I demand what the diameter of another pipe must be to discharge four times as much water in the same time. square by the given proportion, and the square root of the wroduct is se answer. Ruto-Square the given diameter, and multiply o Pros. WI. The sum of any two numbers, and their profucts 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 one 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 losser number. 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. 4428 4-1768 4 times he pro. Then to the 1 sum of 21,5 [numb. +and- 4,5 V81–9 diff of the Greatest n amber, 26,0 4} the diff. | Answers. east number, 17,0 EXTRACTION OF THE CUBE ROOTA 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 quotien, by 300, calling it the divisor. 172 EVOLUTION--R EXTRACTION C F. Roo Is. 5. Seck how often the divisor may be had in the divi Hend, and place the result in the quotient; then multiply the divisor by this last quotient figure, placing the product under the dividend. G. 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 subtrao the subtrahend from the dividend. ---------1. Required the cube root of 18399,744. 18399,744(26,4 Root Ans. 8 2×2–18800–1200) so first dividend |