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 a3 b + 3 a b2 + b2 = 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 a2 b + 3 a b2 + 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.

We have seen above, that when the root is to consist of several figures, the same course is to be pursued as when it consists of only two.

13. What is the third root of 1990865512?

14. What is the third root of 513,345,176,343 ?

15 What is the third root of 217,125,148,004,864 ?

XXXII. The third power of a fraction is found by raising both numerator and denominator to the third power. Thus the third power of is X X =

27

Hence the third root of a fraction is found by finding the third root of both numerator and denominator.

The third of

?

1. What is the third root of

2. What is the third root of

3. What is the third root of 357 = 274 ? 4. What is the third root of 3010?

5. What is the third root of 25 ?

It was remarked with regard to the second root that, when a whole number has not an exact root in whole numbers, its root cannot be exactly found, for no fractional quantity multiplied by itself can produce a whole number. The same is true

with regard to all roots, and for the same reason.

Hence the third root of 25 cannot be found exactly because the numerator has no exact third root. The root of the denominator is 2, that of the numerator is between 2 and 3, nearest to 3. The approximate root is or 11.

6. What is the third root of?

In this, neither the numerator nor the denominator is a per

fect third power; but the denominator may be rendered a perfect third power, without altering the value of the fraction, by multiplying both terms of the fraction by 49, the second power of the denominator.

The root of this is between 4 and 4, nearest to the former. It is evident that the denominator of any fraction may be rendered a perfect third power, by multiplying both its terms by the second power of the denominator. The third root of a whole number which is not a perfect third power, may be approximated by converting the number into a fraction, whose denominator is a perfect third power.

What is the third root of 5?

We may find this root exact within less than of a unit, by converting it into a fraction, whose denominator is the third power of 12.

The root of is between 9 and 1; nearest the latter. The most convenient numbers to multiply by, are the third powers of 10, 100, 1000, &c. in which case, the fractional part of the root will be expressed in decimals, in the same manner as was shown for the second root. The multiplication may be performed at each step of the work. For each decimal to be obtained in the root, three zeros must be annexed to the number, because the third power of 10 is 1000, that of 100, 1000000, &c.

7. The third root of 5 will be found by this method as follows.

The 3d root of 5 is 1.709, within less than Too of a unit. We might approximate much nearer if necessary. The other method explained in the last article may be used if preferred.

8. What is the third root of 173?

The fractional part of this number must first be changed to a decimal.

Hence it appears, that to prepare a number containing decimals, it is necessary that for every decimal place in the root, there should be three decimal places in the power. Therefore we must begin at the place of units, and separate the number both to the right and left into periods of three figures each. If these do not come out even in the decimals, they must be supplied by annexing zeros to the right.

9. What is the approximate third root of 25732.75? 10. What is the approximate third root of 23.1762? 11. What is the approximate third root of 123? 12. What is the approximate third root of 11⁄2? 13. What is the approximate third root of

?

14. What is the approximate third root of ?

XXXIII. Questions producing Pure Equations of the Third Degree.

1. A man wishes to make a cellar, that shall contain 31104 cubic feet; and in such a form, that the breadth shall be twice the depth, and the length 13 the breadth. What must be the length, breadth, and depth?