CUBE ROOT.

193. The Cube Root of a number is one of the three equal factors which produce it; as, 2 is the cube root of 8, because 2 × 2 × 2 = 8.

The relation between the number of places in the cube root of any number, and in the number itself, may be shown by the following

PRINCIPLES.—I. The cube of any number contains three times as many figures, or three times as many less one or two, as the number itself.

II. The cube root of any number contains as many terms as there are periods of three figures each, in the number.

MENTAL

EXERCISES.

1. What is the cube root of 8?

SOLUTION.-To find the cube root of 8, resolve it into its three equal factors, 2 × 2 × 2.

2. What is the cube root of 27? Of 64 ? 125 ? 1000? 8000? 27000?

3. Find the cube root of 1.27. 11. 1000. 4. Find the cube root of 33. 181.

5. What is the side of a cube whose solidity is 512 cu. ft.?

6. If a cubical box contains 216 cu. in., what is the length of one of its sides?

7. Find the side of a cube whose solidity is 219 cu. ft.

8. If a cubical bin contains 343 cu. ft., what is the length of one of its sides?

(3ť2+3t × u+u2) u = 1821 × 9 = 16389

SOLUTION. As the given number contains two periods, its root will consist of two places, tens and units. Hence, t3+3ť2 × u+3tx u2+u3

24389.

The greatest cube in the period 24, is 8; and its root, 2, is the tens figure of the required root. Subtracting the cube of the tens, we get a remainder of 16389, which equals 32 × u +3t × u2 + u3.

312, or the trial divisor, 1200, is contained in 16389, 9 times, and 9 is probably the units' figure of the root.

Then, 312 1200; 3txu = 540; u2 = 81; and the sum = 1200+ 540+81 = 1821, which multiplied by the u, = 1821 × 9 = 16389, the exact remainder. The required root, therefore, is 29.

SOLUTION. In the above operation, instead of 3 times the square of the first term considered as tens, take 300 times the square, which gives the same result; and in completing the divisor, multiply the product of the first and second terms by 30.

To find the second trial divisor, take the square of the first two terms of the root considered as one. In like manner square the first three terms, for the third trial divisor.

RULE.-I. Beginning at units, separate the number into periods of three places each.

II. Having found the root of the left-hand period, subtract its cube from the period, and to the remainder annex the next period for a new dividend.

III. Take 300 times the square of the first term of the root for a trial divisor; divide and take the quotient for the probable second term of the root.

IV. To complete the divisor, multiply the product of the first and second terms of the root by 30; 2dly,

square the second term; and then add both results to the trial divisor.

V. Multiply the complete divisor by the second term, and find the remainder as before; bring down the next period, and thus proceed until all the periods are exhausted.

VI. To find a second, or third trial divisor, take 300 times the square of the entire root thus far found.

NOTES.-1. If the product of the complete divisor by its corresponding term of the root, should exceed the dividend, the term of the root should be diminished.

2. When the dividend does not contain the complete divisor, write a cipher in the root, bring down the next period, annex two ciphers to the trial divisor, and proceed as before.

3. If the number of decimal places in the given number be not divisible by 3, make it so by annexing ciphers.

4. If the given number be not a perfect cube, annex as many periods of ciphers as there are to be decimal places in the root.

GEOMETRICAL METHOD.

What is the length of the edge of a cube containing 300763 cubic feet?

If from the given cube, Fig. 1, we subtract a cube, Fig. 2, whose

edge is 60 ft., there will remain a solid containing 84763 cu. ft., as represented by Fig. 3. This solid consists chiefly of the three rectangular slabs A, B, and C, each of which is 60 ft. long and 60 ft.

wide. It is evident that the thickness of these slabs added to the length of the cube removed, will be equal to the edge of the given cube. Dividing the approximate sum of their volumes, 84733 cu.