6. What is the cube root of 117649?

7. What is the cube root of 350356403707485209 ?

Ans. 49.

when added together, by taking two at a time, produce six distinct sums, each of which is a perfect cube. What are the six roots of these cube numbers?

Ans.

19146344, 21062342, 60097344, 23021160, 60359866, 60571840.

Case II.

To extract the cube root of a decimal fraction, or of a number consisting partly of a whole number, and partly of a decimal value, we have this

RULE.

I. Annex ciphers to the decimals, if necessary, so that the whole number of decimal places may be divisible by 3.

II. Separate the decimals into periods of three figures each, counting from the decimal point towards the right, and proceed as in whole numbers.

NOTE.-If the given number has not an exact root, there will be a remainder after all the periods have been brought down. The process may be continued by annexing ciphers for new periods

EXAMPLES.

1 What is the cube root of 0·469640998917 ?

Ans. 0.7773.

2. What is the cube root of 18·609625?

Ans. 2.65.

3. What is the cube root of 1.25992105?

Ans. 1'08005974, nearly.

To extract the cube root of a vulgar fraction, or mixed number, we have this

RULE.

I. Reduce the fraction, or mixed number, to its simplest fractional form.

II. Extract the cube root of the numerator and denominator separately, if they have exact roots; but when they have not, reduce the fraction to a decimal, and then extract the root by Case II.

When there are many decimal places required in the root, we may, after obtaining one more decimal figure than half the required number, find the rest by dividing the remainder by the last term of the second column.

Before dividing, we can omit from the right of the divisor so many figures as to leave but one more than the number of additional figures required in the root, observing to omit from the right of the dividend one figure less than was omitted in the divisor. The division, must then be performed according to the abridged method, as explained under Art. 41.

EXAMPLES.

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

8

In this example, we proceed in the usual way, until we obtain 191293; the remainder is 12984369243; the last term of the second column is 109777313919; therefore, we obtain four more figures by dividing 12984369243 by 109777313919; but these four figures may be obtained with equal accuracy by dividing 12984 by 10977, which gives the remaining figures

1182.

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

In this example, after obtaining seven decimal figures in the root, by the usual process, the remainder was 29701189129875, and the last term in the second column was 119054974974025; and, since we wish but six figures by division, we reject seven figures from the right of the remainder, and eight figures from the right of the term of the second column, and then divide by the rule