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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 froin 34,965, omituing 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 'vas found in finding its third po:ver.

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 forin the dividend.

llence we chain the following rule for finding the third foot.

Prepare the number by beginning at the right and separating it into parts or periods of three figures each, putting a comina or point between. The lefi han period may consist of one, tico, or three figures.

Find the greatest third power in the left hand period, and write the rool in the place of a quotient. Subtract the power from the period. To the remainder bring do:en 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 augmenteil, to the third power. If this is greater than the first two periods, diminish the quotiert by one or more, until you obtain a third power, which may be subtructed from the first tuo 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 pouer 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 saine 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. Let a = 30 and b = 7; then

(a+b)= (37)' = a + 3 a' b + 3 ab +63 = 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 a' b + 3 a b +63 = b (3 a' + 3 ab + b).

But the 3 a' has already been found for the divisor. • We must now find 3 a b and b'; add all together, and multiply the sum by b, and the third power will be completed.

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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.

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10. What is the third root of 185193 ? ? 11. What is the third root of 8365427 ? 12. What is the third root of 77308776?

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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 ?

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XXXI. 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 = .

Hence the third root of a fraction is found by finding the third root of both numerator and denominator. The third of is = 4.

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It was remarked with regard to the second root that, when a whole number has not an exact root in whole numbers, its root eannot 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 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 14.

6. What is the third root of ? .

In this, neither the numerator nor the denominator is a perfect 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.

3 x 49 – 147
7 x 49 343

The root of this is between 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,

(12): = 1728 5= 1948. The root of 1491 is between 1, and {i ; 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.

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remainder

.008 556 171. The 3d root of 5 is 1.709, within less than dog of a unit. We might approximate much nearer if necessary. The other method explained in the last article may be used if preferred.

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