become the divifor, or true denominator to its own fraetion, which fraction must be annexed to the quotient, to complete the root. Suppose the root to be 12, whien fquared it will be 144, and multiplied by 3, it makes 432, to which add 36, the triple number of the root, and it produces 468 for a denominator. SECOND METHOD. RULE I. Having pointed the given number into periods of three figures each, find the greatest cube in the left hand period, fubtracting it therefrom, and placing its root in the quotient; to the remainder bring down the next period, and call it the dividend. 2. Under this dividend write the triple fquare of the root, fo that units in the latter may ftand under the place of hundreds in the former and under the faid triple fquare, write the triple root, removed one place to the right hand, and call the fum of these the divifor. 3. Seek how often the divifor may be had in the dividend, exclufive of the place of units, and write the refult in the quotient. 4. Under the divifwrite the product of the triple fquare of the root by the last quotient figure, fetting down the unit's place of this line, under the place of tens in the divifor; under this line, write the product of the triple root by the fquare of the last quotient figure, fo as to be removed one place beyond the right hand fig. ure of the former; and, under this line, removed one place forward to the right hand, write down the cube of the laft quotient figure, and call their fum the fubtrahend. 5. Subtract the fubtrahend from the dividend, and to the remainder bring down the next period for a new U "dividend, dividend, with which proceed as before, and so on, till the whole be finished. EXAMPLE. Required the Cube Root of 16194277 ? 16194277(253=Root. Firft dividend. Triple fquare of 2. =3 First divifor. Triple fquare of 2 multiplied by 5. = Cube of 5. Firft fubtrahend. Triple fquare of 25 multiplied by 3. 27 Cube of 3. FIRST METHOD BY APPROXIMATION. RULE 1. Find by trial, a cube near to the given number, and call it the fuppofed cube. 2. Then, 2. Then, as twice the fuppofed cube, added to the given number, is to twice the given number, added to the fuppofed cube, fo is the root of the fuppofed cube, to the true root, or an approximation to it. 3. By taking the cube of the root, thus found, for the fuppofed cube, and repeating the operation, the root will be had to a greater degree of exactness. EXAMPLE. It is required to find the cube root of 54854153 Let 64000000=fuppofed cube, whofe root is 4c0; Then 64000000 128000000 54854153 54854153 109708306 As 182854153: 173708306 :: 400 4C0 182854153)69483321400379=root nearly. Again, let 54439939-fuppofed cube, whofe root is 379 Then, 54439939 108879878 54854153 54854153 109708306 54439939 SECOND METHOD BY APPROXIMATION, RULE. Divide the refolvend by three times the affumed root, and referve the quotient. 2. Subtract one twelfth part of the fquare of the affamed root from the quotient. 4. Extract the fquare foot of the remainder. To this root add one half of the affumed root, and the fum will be the true root, or an approximation to it a Take this approximation as the affumed root, and, by jepeating the procefs, a root farther approximated will be found, which operation may be farther repeated as often as neceflary, and the root difcovered to any affign. ⚫d exactness. NOTE. In order to find the value of the first affumed root, in this or any other power, divide the refolvend into periods, by beginning at the place of units, and including in each period, fo many figures as there are units in the exponent of the root; viz 3 figures in the cube root; 4 for the biquadrate, and fo Then by a table of powers, or otherwife, find a figure, which, (being involved to the power whofe exponent is the fame with that of the required root) is the nearest to the valus of the first period of the refolvend at the left band; and to Det figure anpex fo many cyphers as there are periods remaining in the integral part of the refolvend; this figure, with the cypters annexed, will be the affumed root: And it is of no importance whether the figure thus chofen, be, when involved, greater or lefs than the left hand period. 1. What is the cube root of 436036824287? 7000 affumed root. 8 21,000)436036824287(20763658,2994 Subtr.7000 X 7000÷124083333,3333 No16680324,9661=4084,15 Add the affumed root,=3500 And it gives the approximated root=7584,15. For For the fecond operation, use the approximated root as the affumed one, and proceed as above. THIRD METHOD BY APPROXIMATION. I. Affume the root in the ufual way, then multiply the fquare of the affumed root by 3, and divide the refolvend by this product; to this quotient add 3 of the affamed root, and the fum will be the true root, or an ap proximation to it. 2: For each fucceeding operation let the last approx. imated root be the affumed root, and proceeding in this manner, the root may be extracted to any affigned ex. actness. What is the cube root of 7 ? Let the affumed root be 21 the divifor. Then 2X2 X3=121 12)7,0(,583, to this add of 2=1,333, &c. that is, 583+1,333 1,916 approximated root. Now affume. 1,916 for the root, then, by the fecond 7 procefs, the root is-2+X1,9161,9126, &c.. 3.X 1,916. APPLICATION AND USE OF THE CUBE: ROOT. 1: To find two mean proportionals between any two› given numbers. RULE I. Divide the greater by the lefs, and extract the cube root of the quotient. 2. Multiply the root, fo found, by the leaft of the given numbers, and the product will be the leaft. 3. Multiply this product by the fame root, and it will give the greatest. EXAMPLES. |