dividing this remainder by three times the second power of the tens; for 3 a b divided by 3 a2 gives b. As the other parts however will always be small in comparison with this, if we divide the remainder by three times the second power of the tens, we shall be able to judge very nearly what is the root, and the number of trials will be limited to very few. 30 X 30 900, and 900 x 3 2700 and 15875 divided by 2700 gives 5. I now add the 5 to the root and it becomes 35. To see if this is right, I raise 35 to the third power. 35 × 35 × 35 = 42875, therefore 35 is the true root. 4. What is the third root of 79507? Operation. 79,507 (40+3=43 root. 15,507 (40 x 40 x 3 4800 divisor. 43 × 43 × 43 = 79,507. As the number consists of five places, the power of the tens must be sought in the 79000. The greatest third power in 79000 is 64000, the root of which is 40. I subtract 64000 from 79507 and there remains 15507, which I divide by three times the second power of 40, viz. 4800, and obtain a quotient 3, which I add to 40. I raise 43 to the third power, and find that it gives 79507. If it produced a number larger or smaller, I should put a smaller or larger number in place of 3 and try it again. 5. What is the third root of 357911? It was observed above, that the third power of 10 is 1000, the third power of 100 is 1000000; that of 1000 is 1000000000, &c. That is, the third power of a number consisting of one figure cannot exceed three places; that of a number consisting of two places cannot contain less than 4 places nor more than 6; that of 3 places cannot contain less than 7 nor more than 9 places, &c. Hence we may know immediately of how many places the third root of any given number will consist, by beginning at the right and separating the number into parts of 3 places each. The left hand part will not always contain 3 places. In the present instance, the number 34,965,783, thus divided consists of three parts, therefore the root will contain 3 places or figures. In the formula (a + b)3 = a3 +3 a2 b + 3 a b2 + b3, if we consider a as representing the hundreds of the root, and b the tens and units, we observe that the third power consists of the third power of the hundreds, plus 3 times the second power of the hundreds, multiplied by the units and tens, &c. Hence we shall find the hundreds of the root by finding the highest third power contained in the 34,000,000, and taking its root. The largest third power is 27,000,000, the root of which is 300. Subtracting 27,000,000 from the whole sum, the remainder is 7,965,783. If this contained exactly 3 a2 b, that is, 3 times the second power of the hundreds by the tens and units, the other two figures of the root might be found immediately by division. As it is, it is evident, that it will enable us to judge very nearly what the next figure, or tens, of the root must be, and its correctness must be proved by trial. 300 X 300 X 3 270000. 7,965,783 divided by 270000 gives for the first figure of the quotient 2, which being the tens is 20. This added to the root already found makes 320. If in the above formula, we consider a as representing the hundreds and tens instead of the hundreds; and b as representing the units; it shows us that the power contains the third power of the hundreds and tens, plus 3 times the second power of the hundreds and tens multiplied by the units, &c. În the present instance a 320. If now we subtract the third power of 320 from the whole sum, viz. 34,965,783, and divide the remainder by 3 times the second power of 320, we shall find the other figure, or units, of the root. When we have raised 320 to the third power, we can ascertain whether the second figure, 2 is right. This subtracted from 34965783 leaves 2197783. 320 X 320 × 3 = 307200. 2197783 being divided by 307200 gives a quotient 7. This added to 320 gives 327 for the root. 327 X 327 X 327 34,965,783. Therefore the result is correct. If the root consists of four or more places, the same mode of reasoning may be pursued by making a first equal to the highest figure in the root, and b equal to all below, until the second figure of the root is obtained, and then making a equal to the two figures already obtained, and b equal to the rest, and so on. The work may be considerably abridged by omitting the zeros in the work, and also the numbers under which they fall. The work of the above example will stand thus. As the third power of hundreds can have no significant figure below 1000000, and as the third power of 300 and 3 have the same significant figures, I raise 3 to the 3d power and subtract it from 34, as if it stood alone. Then, to form the divisor, hundreds are multiplied by hundreds, therefore there can be no significant figure below 10000. And it being the tens of the root that are to be found, it is sufficient to bring down one figure of the next period to form the dividend. Having found the second figure of the root, I raise 32 to the third power, and subtract it from 34,965, omitting the last period, because the third power of the tens can have no significant figure below 1000. To form the second divisor I multiply the second power of 32 by 3. For the dividend, it is sufficient to bring down one figure of the last period to the right of the remainder, because the divisor, being tens, multiplied by tens, can have no significant figure below 100. Note. The second power of the 32 was found in finding its third power. If it happens that the divisor is not contained in the dividend, a zero must be put in the root, and then the next figure must be brought down to form the dividend. Ilence we obtain the following rule for finding the third oot. Prepare the number by beginning at the right and separating it into parts or periods of three figures each, putting a comma or point between. The left hand period may consist of one, two, or three figures. Find the greatest third power in the left hand period, and write the root in the place of a quotient. Subtract the power from the period. To the remainder bring down the first figure of the next period for a dividend. Multiply the second power of the root already found by three, to form a divisor. See how many times the divisor is contained in the dividend, and write the result in the root. Raise the root, thus augmented, to the third power. If this is greater than the first two periods, diminish the quotient by one or more, until you obtain a third power, which may be subtracted from the first two periods. Perform the subtraction, and to the right of the remainder bring down the first figure of the next period to form a dividend and divide it by three times the second power of the two figures of the root, and write the quotient in the root. Then raise the whole root so found, to the third power; and if it is not too large, subtract it from the first three periods; if it is too large, diminish the root as before. To the remainder bring down the first figure of the fourth period, and perform the same series of operations as before. If at any time it should happen that the dividend, prepared as above, does not contain the divisor, a zero must be placed in the root, and the next figure brought down to form the dividend. We explained a method in the extraction of the second root, more expeditious than to raise the root to the second power every time a new figure is obtained in the root. A similar method may be found for the third root, though it is rather difficult to be remembered. (a + b)3 = (37)3 = a3 + 3 a2 b + 3 a b2 + b3 = 50653 To find the third root of 50653, find the first figure of the root as explained above. Then form the divisor as above, and find the second figure of the root. Then instead of raising the whole to the third power, it may be completed from the work already done. The third power of the first figure being found and subtracted, the remaining part is 3 a2b+3ab2 + b3 = b (3 a2 + 3 a b + b2). But the 3 a has already been found for the divisor. We must now find 3 a b and b2; add all together, and multiply the sum by b, and the third power will be completed. |