2. What is the square root of 1296 ? 56644.? 5499025 ? 5. Of 36372961 ? 6. Of 184,2 ? 7. Of 9712,693809? 8. Of 0,45369? 9. Of 2002916 ? 10, Of 45? ANSWERs. 36 23,8 2345 6031 13,57 + 98,553 ,673+ ,054 6,708 TO EXTRACT THE SQUARE ROOT OF VUL GAR FRACTIONS. 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 numeratur, and the root of the denominator, for a new denominator, 2. If the fraction be a surd, reduce it to a decimal, and y extract its root. EXAMPLES. 152 SURDS. ANSWERS. 1 32 6. What is the square root of 35? 9128+ 7. What is the square root of 4? ,7745+ 8. Required the square root of 361 ? 6,0207+ APPLICATION AND USE OE THE SQUARE ROOT. PROFLEMI.-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. V5184=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 i!ic mean proportional between 18 and 72 ? 72 x 18=1296, and ✓ 1296=36 Ans. Pror. IV. To form any body of soldiers so that they may Le doulile, 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,whicli double, triple, &c. and the product will be the 1.umber ia raok. EXAMPLES. Let 13122 men be so formed, as that the number in rank may lie double the number in file. 13122 -2=6501, and 76561=81 in file,' and 81 x 2 162 in rank. Pror. V. Adinit 10 hhds. of water are discharged Through a leaden pipe of 2 inches in diameter, in a certain time; I demand what the diameter of another pipe must be to discharge four times as inuch water in the same time. Rule.-Square the given diameter, and multiply said square by the given proportion, and the square root of the product is the answer. 22,5, and 2,5x2,5=6,25 square. 4 given proportion. 12:5,09-5 inclı. diam. Ans. Prob. VI. 'The sum of any two numbers, and their prolucts 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 ine said difference added to half the sun, gives the greater of the two numbers, and the said hali difference subtracted from the half sum, gives the lesser auber. EXAMPLES. The sum of two numbers is 43, and their product is 1?; what are those two numbers? The suin of the numb. 43 x 13=1849 square of do. The product of do. 442 x 4=1703 4 times the pin. Then to the sum of 21,5 [numb. tand V8129 din. of the Greatest n.mber, 4. the dili 26,0 Answers. 17,0) east number, 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 giveni Dininka ber. RULE. 1. Separate the given number into periods of three figitress 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 sail 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 divi 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; tlien 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 inanner, 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 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. 18399,744(26,4 Root. Ans. 8 2x2=4x300=1200)10399 first dividend. 7200 6*6=36x2=72x30=2160 6x6*6= 216 9576 Ist subtrahend. 26:< 26=676 X 300202800)823744 2d dividerrd. 811200 4*4=16 x 26=416 x 30= 12480 4x4X4 64 323744 2d subtrahend. Note.--The foregoing example gives a perfect root; and if, when all the periods are exhausted, there happens to 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 ? 52,7 6. Of 29,508381 ? 3,09+ 7. Of 80,763 ? 4,32+ 8 Of ,162771336 ? ,546 9. Of ,000684134 ? ,088+ 10. Of 1226153:27232? 4968 4. Of RULE. 1. Find by trial, a cube near to the given number, and call is ihe supposed cube. 2. Then, as twice the supposed cubo, added to the given number, in 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 cuhe, and repeating the operation, the root will be had to a groater 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--1,3 x 1,3 x 1,3=2,197=supposed cube. Then, 2,197 2,000 given number. 2 2 : : As 6,394 1,3 : 1,259) foot, which is true to the last place of decimals; but night by teos peating the operation, be brought to greater exactress. 2. What is the cube root of 581,277056 ? Ans. Dy3. |