« ΠροηγούμενηΣυνέχεια »
the servants ; and the whole sum of money taken out was $1458. How many gentlemen were there ?
Ans. 9 gentlemen. 3. A poulterer bought a certain number of fowls. The first year each fowl had a number of chickens equal to the original number of fowls. He then sold the old ones.
The next year each of the young ones had a number of chickens equal to once and one half the number which he first bought. The whole number of chickens the second year was 768. What was the number of fowls purchased at first?
It appears that in equations of the third degree, as in those of the second degree, the power of the unknown quantity must first be separated from the known quantities, and made to stand alone in one member of the equation, by the same rules as the unknown quantity itself is separated in simple equations When this is done, the first power or the root must be found, and the work is finished.
Extraction of the Third Root.
The third power of a quantity is easily found by multiplication, but to return from the power to the root, is not so easy. It must be done by trial, in a manner analogous to that employed for the root of the second power.
We shall hereafter have occasion to speak of the root of the fourth power, of the fifth power, &c. In order to distinguish them the more readily, we shall call the root of the second power, the second root of the quantity ; that of the third power, the third root, that of the fourth power, the fourth root, &c. To preserve the analogy, we shall sometimes call the root of the first power, the first root.
N. B. The first power, and the first root, are the same thing, and the same as the quantity itself.
It always has been, and is still the practice of mathematicians, to call the second root the square root, and the third root the cube root, and sometimes, though not so universally, the fourth root the bi-quadrate root. But as these terms are unappropriate, they will not be used in this treatise.
When the root consists of but one figure, it must be found by trial. When the root consists of more than one place, it
must still be found by trial, but rules may be made, which will reduce the number of trials to very few, as has been done above for the second root.
In order to find the rules for extracting the third root, it will be necessary to observe how the third power is formed from the first, when the first consists of several figures.
Let a = 30 and b = 5; then a + b = 35.
a = 30 x 30 x 30 = 27000
Hence it appears, that the third power of a number consisting of units and tens, contains the third power of the tens, plus three times the second power of the tens multiplied by the units, plus three times the tens multiplied by the second power of the units, plus the third power of the units.
Farther, the third power of 10, which is the smallest number with two places, is 1000, which consists of four places; and the third power of 100, is 1000000, which consists of seven places. Hence the third power of tens will never be less than 1000, nor so much as 1000000.
If, therefore, there are tens in the root, their power will not be found below the fourth place; and if the root consists of tens without units, there will be no significant figure below 1000.
To trace back again the number 42875, the root of the tens will be found in the 42000, and this must be found by trial.
30 X 30 X 30 = 27000, and 40 x 40 x 40 = 64000. The largest third power in 42000 is 27000, the root of which is 30. Now ) subtract 27000 from 42875, and the remainder is 15875, which contains the product of three times the second power of the tens by the units, plus, &c. If it contained exactly three times the second power of the tens multiplied by the units, the units of the root would be found immediately by
dividing this remainder by three times the second power of the tens; for 3 a b divided by 3 a' 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 X 35 x 35 = 42875, therefore 35 is the true root.
4. What is the third root of 79507 ?
79,507 (40' + 3 = 43 root.
15,507 (40 X 40 X 3 = 4800 divisor. 43 X 43 X 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)= a + 3 a b + 3 abi + b', 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 upits 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 a' 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 mụst 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. In 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.
320 x 320 X 320 = 32768000. This subtracted from 34965783 leaves 2197783.
320 x 320 X 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.
Root. 34,965,783 (300 + 20 +7= 327. - 27,000,000 3d power of 300
1st divisor 1st divid. 7,965,783 (270,000
300 x 300 x 3 - 32,768,000 3d power of 320
2d divisor 2d divid. 2,197,783 (307,200
320 X 320 X 3 34,965,783 = 3d power of 327. The same without the zeros.
34,965,783 (327 3d power of 3 27
(27 1st divisor = 3 X 3
Ist dividend 7,9
of 32 32 768
2 197,7 (2072 34,965,783.
(32)* X 3