Examples.

1. What is the cube root of 7 carried to 9 decimal places?

In this example we proceed in the usual way, until we obtain 1.91293, the remainder was 12984369243; the last term of the second column was 109777313919; therefore we must obtain 4 more figures by dividing 12984369243 by 109777313919; but since we wish but 4 more figures, they may be obtained with equal accuracy by dividing 12984 by 10977, which gives the remaining figures 1182.

2. Extract the cube root of 2=0.25 to 13 decimal places.

In this example, after obtaining 7 decimal figures in the root by the usual process, the remainder was 29701189129875, and the last term in the 2d column was 119054974974025; and since we wish but 6 figures by division, we reject 7 figures from the right of the remainder, and 8 figures from the right of the term of the second column; and then divide by

the rule for abridging the work, and obtain the remaining figures of the root.

3. Extract the cube root of 9 to 9 decimals.

4. What is the cube root of 15 to 5 decimal places?

Ans. 2.50222.

5. What is the cube root of 3.375 to 8 decimals?

Ans. 0.68278406.

6. What is the cube root of 0.0000031502374 to 13 decimals?

Ans. 0.0146593402919.

7. What is the cube root of to 21 decimals?

Ans. 0.793700525984099737376..

8. What is the cube root of to 10 decimals?

9. What is the cube root of 14 to 7 decimals?

10. What is the cube root of to 8 decimals.

365

EXAMPLES INVOLVING THE PRINCIPLES OF THE CUBE ROOT.

80. It is an established theorem of geometry, that all similar solids are to each other as the cubes of their like dimensions.

1. If a cannon ball, 3 inches in diameter, weighs 8 pounds, what will a ball of the same metal weigh, whose diameter is 4 inches?

By the above theorem, we have 33: 43: 8 pounds : 183 pounds, for the answer.

2. What is the side of a cube, which will contain as much as a chest 8 feet 3 inches long, 3 feet wide, and 2 feet 7 inches deep?

3. Suppose the diameter of the sun is 887681 miles, the diameter of the earth 7912 miles, how many times greater in bulk is the sun than the earth?

Ans. 1412251 times, nearly. 4. Suppose the diameter of the moon to be 2160 miles, how many times greater in bulk is the sun than the moon?

5. How many cubic quarter inches can be made out of a cubic inch? Ans. 64.

6. Required the dimensions of a rectangular box, which shall contain 20000 solid inches, the length, breadth, and depth, being to each other as 4, 3, and 2.

7. Four ladies purchased a ball of exceeding fine thread, 3 inches in diameter. What portion of the diameter must each wind off so as to share of the thread equally?

- SOLUTION.

After the first one had wound off her share, the ball which

remained would contain as much thread as it did in the first

3

3

place. Therefore its diameter was 3√3√6=2.72568 inches, nearly.

The diameter, after the second one had wound off her share,

3

was 34-2.38110 inches, nearly.

The diameter, after the third one had wound off her share, was 3√√21.88988 inches, nearly.

Hence, the portions of the diameter which they must wind off are as follows:

ROOTS OF ALL POWERS.

81. Whenever the index, denoting the root required, is a composite number, the root can be found by successive extractions of the roots denoted by the prime factors of the original index.

Thus, the 4th root can be found by extracting the 2d root twice in succession.

The 6th root is obtained by extracting the 3d root of the 2d root.

The 8th root is found by extracting the 2d root three times in succession..

When the index denoting the root is a prime, we must have some direct method of obtaining the root.