7. Subtract the subtrahend from the dividend, and to the remainder bring down the next period for a new dividend, with which proceed as before, and so on, till the whole is completed.

NOTE. The same rule must be observed for continuing the operation, and pointing for decimals, as in the square root.

ILLUSTRATION.

We suppose we have 46,656 cubic blocks of granite, which measure one foot on each side. With these we wish to erect a cubical monument. It is required to ascertain how many blocks, or feet, will be the length of one side of the monument. It is evident that the number of blocks will be equal to the cube root of 46,656. As the given number consists of five figures, its cube root will contain two places; for the cube of any number can never contain more than three times that number, and at least but two less. We therefore separate the given number into periods of three figures each, putting a point over the unit figure, and every third figure beyond the place of units; thus, 46,656. We find by the table of powers, or by trial, the greatest power in the left-hand period, 46 (thousand), is 27 (thousand), the root of which is 3. This root we write in the quotient; and, as it will occupy the place of tens, its real value is 30. If this be considered the side of a cube, it will contain 27,000 cubic feet, 30 × 30 × 30 = 27,000 feet.

Let this cube be represented by figure 1, each of whose sides measures 30 feet; therefore its contents will be 30 × 30 × 30 = 27,000 feet, as above. We subtract the contents of this cube from 46,656, and there remain 19656 cubic feet.

Fig. 1.

OPERATION.

46,656(36 Ans.
27,000

2700) 19,656

16,200

3,240

216 19,656

Or we might have subtracted the cube of 3, = 27, from the first period, and to the remainder have brought down the next period, and the result would have been the same. (See operation.) The cubic blocks that remain must be applied to the three sides of figure 1. For, unless a cube

be equally increased on three sides, it ceases to be a cube. To effect this, we must find the superficial contents of three sides of the cube, and with these we must divide the remaining number of cubic feet or blocks, and the quotient will show the thickness of the additions. As the length of a side is 30 feet, the superficial contents will be 30 X 30= 900 square feet, and this multiplied by 3, the number of sides, will be 900 X 32700 feet. With this as a divisor, we inquire how many times it is contained in 19,656, and find it to be 6 times (one or two units, and sometimes three, must be allowed on account of the other deficiencies in enlarging the cube). This 6 is the thickness of the additions to be made to the three sides of the cube, and by multiplying their superficial contents by it, we have the solid contents of the additions to be made, 2700 X 616200; that is, we multiply the triple square by the last quotient figure, and this may be represented by the three superficies ABCD, EFGH, and I KL M. (See figure 2.)

Having applied these additions to our cube, we find there are three other deficiencies, a b c d, e f g h i j k l, the length of which is equal to that of the additions, 30 feet, and the height and breadth of each are equal to the thickness of the additions, 6 feet. To find the contents of these, we multiply the product of their length, breadth, and thickness by their number; thus, 6 × 6 × 30 × 3 = 3240; or, which is the same thing, we multiply the triple quotient by the square of the last quotient figure; thus, 90 × 6 × 6=3240. See rule.

Having made these additions to the cube, we still find one other deficiency, N. (See figure 2) The length, breadth, and thickness of which are equal to the thickness of the former additions, viz. 6 feet. The contents of this are found by multiplying its length, breadth, and thickness together; that is, cubing the last quotient figure; thus, 6 × 6 ×6=216. By making this last addition, we find that our cubical monument is finished, and that the

d

D

a

Fig. 2.

H

G

k

E

F

i

b

12/K

N

I

Bh

L

M

first figure together with the several additions is equal to the cubical blocks, 46,656.

46656 contents of the whole monument.

1. Required the cube root of 77308776.

32207762d subtrahend. 1260 × 6 × 6= 45360

6×6×6= 216

2d subtrahend = 3220776

2. What is the cube root of 34965783 ? 3. What is the cube root of 436036824287? 4. What is the cube root of 84.604519 ? 5. Required the cube root of 54439939. 6. Extract the cube root of 60236288. 7. Extract the cube root of 109215352. 8. What is the cube root of 116.930169 ? 9. What is the cube root of .726572699 ? 10. Required the cube root of 2. 11. Find the cube root of 11.

Ans. 327. Ans. 7583. Ans. 4.39. Ans. 379. Ans. 392.*

Ans. 478. Ans. 4.89. Ans. .899. Ans. 1.2599+. Ans. 2.2239

12. What is the cube root of 122615327232? Ans. 4968.

13. What is the cube root of 12? 14. What is the cube root of 1831? 15. What is the cube root of $1? 16. What is the cube root of 593

512

6892

Ans.. Ans. 14.

Ans. 8. Ans..

To find the cube root of any number mentally, less than 1,000,000, when the number has an exact root.

RULE. - As there will be two figures in the root, the first may easily be found mentally, or by the table of powers; and if the unit figure of the power be 1, the unit figure in the root will be 1; and if it be 8, the root will be 2; and if 7 it will be 3; and if the unit of the power be 6, the unit of the root will be 6; and if 5, it will be 5; if 3, it will be 7; if 2, it will be 8; and if the unit of the power be 9, the unit of the root will be 9. This will appear evident by inspecting the table of powers.

17. What is the cube root of 97336 ?

Ans. 46.

Explanation. By examining the left-hand period, we find the root of 97 is 4, and the cube of 4 is 64. The root cannot be 5, because the cube of 5 is 125. The unit of the power is 6; therefore, by the above rule, the unit figure in the root is 6. The answer, therefore, is 46.

18. What is the cube root of 132651 ? 19. What is the cube root of 148877 ? 20. What is the cube root of 175616? 21. What is the cube root of 185193? 22. What is the cube root of 238328 ? 23. What is the cube root of 262144? 24. What is the cube root of 389017 ? 25. What is the cube root of 405224 ? 26. What is the cube root of 531441 ? 27. What is the cube root of 24389 ? 28. What is the cube root of 42875 ?

Ans. 51.

Ans.

Ans.

Ans.

Ans.

Ans.

Ans.

Ans.

Ans.
Ans.

Ans.

SECOND METHOD OF EXTRACTING THE CUBE ROOT.

The following rule for the extraction of the root of the third power, though it is essentially the same with the former, may yet serve to make the reasons for the several steps of the operation more intelligible to the learner.

RULE. Separate the number whose root is to be found into periods, as under the former rule, and find by trial the greatest root in the lefthand period, and put it in the place of the quotient.

Subtract the third power of this root from the period to which it belongs, and to the remainder bring down the next period for a dividend.

Then, to find a divisor, annex a cipher to the root already found, and multiply twice the number thus formed by the number itself, and to the product add the second power of this number.*

*This is the same as multiplying the square of the radical figure by 300, as in the former rule.

Ascertain how many times this divisor is contained in the dividend, and write the result in the quotient.*

Then, to find the subtrahend, multiply this divisor by its quotient, and write the product under the dividend. To this add three times the preceding radical figure with a cipher annexed,† multiplied by the second power of the figure last obtained, and also the third power of this last figure. Subtract the sum of their several products from the dividend above them, and to the remainder bring down the next period for a new dividend. With the parts of the root already found proceed to find a divisor and subtract as above, and so on, till the successive figures of the root are all obtained.

The rationale of the above rule may be made to appear by the solution of the following question.

Let it be required to find the cube root of 17576.

We now raise the quantity 20 +6 to the third power.

20+6 20+6

400+ 120

120 + 36

400+240 + 36
20+ 6

8000+4800+720

2400+1440 +216

8000+7200+2160+21617576.

*This quotient figure must sometimes be less than the one indicated by the divisor, and in extreme cases the divisor may give a quotient too large by several units. The quotient required can, of course, never exceed 9.

This accounts for multiplying by 30, in the foregoing rule, which is a factor in the triple quotient in finding the subtrahend.