444. To find the Cube Root of a number is to resolve it into three equal factors; that is, to find a number which, taken three times as a factor, will produce the given number.

Numbers, 1,
Cubes,

10, 1, 1000,

100, 1000000,

1000.

1000000000.

A

445. Comparing the numbers above with their cubes, we see that the cube of any integral number less than 10 has less than three figures; the cube of any integral number less than 100 and more than 9 has less than seven and more than three figures; and so on. That is, the cube of a number consists of three times as many figures as the root, or of one or two less than three times as many.

446. Hence, to find the number of figures in the cube root of a given number,

Begin at the right, and mark off the number into periods of three figures each, and there will be one figure in the root for each period of three figures in the cube, and if there are one or two figures besides full periods in the cube, there will be a figure in the root for this part of a period.

(For the Algebraic explanation of cube root, see Appendix, page 363.)

447. Written Exercises.

43. What is the length of one side of a cubical block that con

cube in 74 (thousands) is 64 (thousands); therefore, the tens figure of

D

If we know the volume of a rectangular prism and two of its dimensions, we can find the other dimension by dividing the volume by the product of the other two dimensions (Art. 211). Now we know the area of the face B D, viz. 40 x 40 sq. in. = 1600 sq. in. Therefore we know the area of the three adjacent faces, viz. 1600 × 3 sq. in. = 4800 sq. in. This is sufficiently near the area to be covered by the additions to serve as a trial divisor. 10088 4800 = 2+. If now we place upon the three adjacent faces the three additions, each 40 x 40 x 2, we shall find that to complete the cube there are lacking three corner-pieces, each 40 inches in length, 2 in breadth, and 2 in thickness, — that is, the length of each is the linear dimension of the 40 original cube, and the other two dimensions are the same as the thick

ness of the three additions. There

40

fore, the area to be covered by these three corner-pieces is 40 X 2 X =240 square inches. Plac

3 sq. in.

=

ing these three corner-pieces in posi-
tion, there still lacks, to complete the
cube, a little cube whose linear di-
mension is 2 inches, the same as the 2
thickness of the additions; and the
area to be covered by this little cube
is 2 x 2 square inches
= 4 square
inches.

The total area to be covered

by the additions necessary to keep

the cubical form are, therefore,

5044 sq. in.

2

40

02 40

has been found.

40

2

In this figure ABCD represents the surface of the three additions that form the trial divisor, BEFC the first addition to the trial

divisor, and G H KF the second addition; and AEG HKD represents the surface of all the additions that form the true divisor.

NOTE. We do not divide 10088 cubic inches by 4800 square inches. If the thickness of the additions is 1 inch, the cubic inches required to make the addi

tions to the three faces will be 4800 cubic inches; the additions therefore will be as many inches thick as 4800 cubic inches is contained times in 10088 cubic inches.

448. If there are more than two figures in the root, the same reasoning applies. Hence, to find the cube root of a number,

Rule.

1. Beginning at units, separate the number into periods of three figures each.

2. Find by trial the greatest cube in the left-hand period, write its root at the right of the given number, subtract the cube from the left-hand period, and to the remainder annex the next period for a dividend.

3. Square the root figure, annex two ciphers, and multiply this result by three for a trial divisor; divide the dividend by the trial divisor and write the quotient as the next figure of the root.

4. Multiply this root figure by the part of the root previously obtained; annex one cipher and multiply this result by three; add the last product and the square of the last root figure to the trial divisor, and the sum will be the true divisor.

5. Multiply the true divisor by the last root figure, subtract the product from the dividend, and to the remainder annex the next period for a new dividend.

6. Find a new trial divisor, and proceed as before, until all the periods have been employed.

NOTE.

The notes in Art. 442, with slight modifications, are equally applicable in cube root.

2d Trial Divisor 2702 × 3 = 218700 1341576

.270 × 6 × 3 = 4860

2d True Divisor

62=

36

223596 1341576

The first trial divisor is contained 10 times in the dividend, yet the root figure is only 7. The true root figure can never exceed 9, and must in all cases be found by trial.

45. What is the cube root of 67917312?

NOTE 2. Prepare fractions and mixed numbers as in square root (Art. 443).

What is the value of the following expressions: