Ex. 3. The product of three equal factors is 6372783864, which are they?

Therefore each of the three factors = 1854.

which is 185436372783864.

The proof of

340. We shall now show another method of extracting the cube root, which is preferable on account of its brevity.

Taking Example 2 over again, we have :—

Here the number is divided into periods as before, and the cube of 2 is subtracted. To the remain-
der, 9, the next period, 173, is subjoined. The triple of the first figure of the root, 2, is set in column I;
then multiplying this triple by the figure in the root, the product 12 is placed in column II.

With 12, as trial divisor, divide 91 (the two last figures of the remainder being omitted), the quo-
tient is 5, set it as the second figure of the root, and also in column I, after 6, making 65; now multiply
65 by 5, and the product is set in column II, under 12, extending two places of figures more to the
right, and 1525, the sum of these two numbers, is the corrected divisor; this sum is multiplied by the
second figure in the root, and the product 7625 is subtracted; the last period is brought down to the
remainder, and the number we have is 1548512.

Now we must repeat the same process to obtain the third figure of the root as we did for the second,
viz., triple the root, which gives 75, and set the product in column I.; now set the square of 5, the second
figure of the root, in column II., and add the three last numbers, those enclosed by the brackets, the sum
1875 is the trial divisor, which is found to be contained in 15485 (the two last figures being omitted)
8 times; then proceed as before.

We have appended to each line the quantities we have made use of before, in order to render the process intelligible, and after what has been previously said this needs no explanation.

Our third example worked after this method is here represented :

If a number be not a perfect cube, its root is extracted approximately, by adding as many periods of three ciphers as there are decimal figures required in the root.

341. After what has been said about the extraction of the square root of fractions and the extraction of the cube root of numbers, as well as the observations made upon the formation of the cube of fractions, no explanation is required to elucidate the following cases of the extraction of the cube root of both decimal and vulgar fractions.

Instead of reducing to a decimal, we might have multiplied both numerator and denominator by the square of the denominator, and the denominator becomes thus a perfect cube; for

1. Find the cubes of 548, 74302, 129458.

2. Express the cubes of,, 123, 3117.

3. What are the cubes of .3, 2.006, .004, 34.005?
4. Extract the cube roots of 512, 140608, 11089567.
5. Determine the cube roots of 8

1259

T331

33, 13483, 162,378 6. What are the approximate cube roots of 50, .653, 4258, .28, .00459, 037, 1587.962, 00000943.

7. What number is that whose cube root decreased by 3 is equal to 14?

8. In a train of 25 carriages there are 25 persons in each carriage, and each person carries £25, what sum of money do these people possess?

9. What number is that whose half, third, and fourth, multiplied together, the product is 9?

10. Find a number whose third multiplied by its square the product is 1944.

11. The third part of the cube of a number is 171307467, find that number.

12. When 137 is added to the cube of a number, the sum is 2334, what is the number?

13. What is the contents of a reservoir, 30.44 yards long, 20.21 yards broad, and 3.4 yards deep?

14. The product of a number by its square is 4.096, find the number.

15. What is the side of a cubical mound that will be equivalent to one which is 288 feet long, 216 feet broad, and 48 feet high?

16. What is the cube root of the square root of 346, correct to two decimal places?