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inches, let the base be a square, each side 12 inches; the square will contain 144 square inches, and for every inch in height 144 solid inches, and 12 in height 1728 solid inches; that is

12 x 12 x 12 = 1728. The cube of 1 is 1, and of 2 it is 8.

1x1x1=1, or 2 x 2 x 2 = 8, and the cube of 12 is 1728 increased three figures.

9 x 9 x 9 = 729, and 99 x 99 x 99 = 970299. The cube of the largest digit has three places of figures, and the cube of two places of the largest digits has six places of figures; therefore, the increase for one figure in the root is three in the cube; hence the pointing off in periods of three figures.

Extracting the cube root consists in having given the cube or solid contents to find a side.

REM.-As the cube is the product of three dimensions, and as the root is one of those dimensions, the divisor must necessarily have two dimensions. As 11 x 11 x 11 = 1331 = (10 + 1)9 = 1331.

Literal a = 10, b = 1.
a +6
a +5
al + ab

+ ab + 12
a+ 2ab +62
a +5
as + 2a2b + ab?

a b + 2ab2 +33
a: + 3a2b+ 3ab2 +62 ( a to

a3
3a? + 3ab + 32 ) 3a2b + 3ab2 +63

3a2b + 3ab2 +63

COR.-The first term of the root is a, the cube of which is a3; the second term of the root is b, and the required divisor must be contained b times in what remains after deducting as; therefore it must be 3a + 3ab + b2; that is, 3 times the square of the first term of the root, three times the product of the last term and the previous root, plus the square of the last term of the root; by extending the exemplification it is only necessary to repeat the foregoing, as each successive divisor is formed in the same way.

PROBLEMS. 1. Extract the cube root of 1331, which power is thus found, 11 x 11 x 11. Point off in periods of three figures; the root of the left-hand period is 1 ; cube it and subtract; the value of this 1 is 10, when a figure is annexed; hence, 3a? = 10 x 10 x 3 = 300

1,331 ( 11 3ab = 10 x 1 x 3 = 30

1 1 x 1 1

331 331

331

32 =

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2. Extract the root of 970299; point in periods. The root of the first period is 9; cube it and subtract, then bring down the next period and form a divisor ; thus,

3a? = 90 x 90 x 3 = 24300 970,299 ( 99
3ab = 90 X 9 X 3 2430 729
9 x 9 =

241 299
26811 241 299

=

[ocr errors]

81

REM.- When there is one figure in the root and a second figure is taken into consideration, then the first figure must be regarded

as tens, and a zero put after it ; this must be done to the second when a third is considered.

COR.-In order to extract the cube root of a number, point off the number in periods of three figures each, beginning with units; the left-hand period will have one, two, or three, depending on the number of figures. Next, place in the root the largest figure whose cube is either equal to or less than the number in the left-hand period regarded as units; cube the root thus obtained and subtract its cube from the left-hand period, and to the remainder, if there is any, annex the next period ; if there is no remainder, then the next period will be the dividend, and for a trial divisor take three times the square of the root already found regarded as tens; find how often it is likely to be contained in the dividend, after making an allowance for what is to be added to the divisor; annex this figure to the root already found, and add to the trial divisor three times the product of the last figure and the previous root regarded as tens, and to this sum add the square of the last figure of the root; this renders the divisor complete; perform the division, and if there are other periods, bring them down in order and repeat the foregoing process until the entire root is found.

EXAMPLES.

1. Extract the cube root of 1367631.
2. Extract the cube root of 997002999.
3. Extract the cube root of 91125.
4. Extract the cube root of 9.261.
5. Extract the cube root of 4.

Ans. 111.
Ans. 999.

Ans. 45.
Ans. 2.1.

Ans.

6. Extract the cube root of 3

Ans. 17. Extract the cube root of 2515456. Ans. 136. 8. Extract the cube root of 1124864. Ans. 104. 9. Extract the cube root of 8024024008.

Ans. 2002. 10. Extract the cube root of 65939.264. Ans. 40.4. 11. Extract the cube root of .000343. Ans. .07. 12. Extract the cube root of 5.

Ans. 13. 13. Extract the cube root of f.

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REM.-1. The volumes of cubical figures are to each other as the cubes of their edges.

2. In order to extract the cube root of a fraction, first render the denominator a perfect cube.

[blocks in formation]

1. What is the depth of a measure of 1 bushel U. S. dry measure, in the form of a cube? Of 1 bushel ?

Ans. Of a bushel 12.907 + in., of } bushel 10.244+ in.

2. What would be the length of a cubical pile of stone equal in volume to a rectangular pile whose length is 6+ feet, breadth 27 feet, and 8 feet high ?

Ans. 24 feet. 3. If a ball 4 inches in diameter weighs 10 lb., what is the diameter of a ball weighing 640 lb. ?

Ans. 16 inches.

REM.—The volume of spheres are to each other as the cubes of their diameters.

4. What are the dimensions of a cube containing 1728 cu. in. ?

Ans. 12 in.

It is demonstrated in Geometry that similar cubical bodies are to each other as the cubes of their like dimensions; as, if a cube of 1 inch weigh 2 lbs., one of three inches would weigh 27 x 2 = 54 lbs.

5. How many cubes whose edges measure 4 inch, would be contained in a cubical block whose edges are 2 inches ?

1 x 1 x 1 =

64. 2 x 2 x 2 = 8 x 64

= 512, Ans.

6. How many shot, to inch in diameter, can be made of a globe of lead 4 inches in diameter ?

to X to x 16 = 2006
4 X 4 X 4 = 64 x. 4906 = 252144, Ans.

7. How many cubes whose edges measure } in. is contained in a cubical block whose edges are 3 inches ?

Ans. 1729 cubes.

8. How many square feet in the surface of a cube whose yolume is 3375 cubic feet?

Ans. 1350 sq. ft.

9. What is the length of an edge of a cubical bin which contains 500 bushels of wheat ?

Ans.. 8 feet 6 inches.

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