CASE I.

Art. 440. To extract the cube root of a whole number,

or of a pure or mixed decimal.

Ex. 1. What is the cube root of 12167?

WRITTEN PROCESS.

Ans. 23.

EXPLANATION.

12167(23

By Art. 428, this number has a cube root of two figures, namely, units and tens. The three right-hand figures 167, must contain the cube of the

units of the root; the other two figures, 12, must contain the cube of the tens. Therefore, in general, each group of three figures in order toward the left must furnish a figure of the cube root. Hence, mark off the number from units, inclusive, into groups of three figures. By Art. 430, the two periods of this number, 12167, must

contain

1. The cube of the tens of the root.

2. Three times the square of its tens its units.
3. Three times its tens X its units X its units.
4. The cube of its units;

the sq. of its units

its units.

The greatest cube in the second period, 12, is 8, whose root is 2, which is the tens of the whole root. Taking 8 from 12, the remainder of the entire number is 4167, from which the units of the root must be obtained. Now, this 4167 must contain the last three of the above mentioned products, of each of which the units are one factor. Hence, dividing by one of the other factors, namely, three times the square of the tens, will probably give the other factor, namely, the units. On filling out the required products, they do not amount to more than the dividend: hence 3 is the units of the root. If the products filled out should in any case amount to more than the dividend, we must try a less figure for the root. In the foregoing written process we delayed multiplying by the units till we found the sum of the other factors, namely, 1389, thus multiplying them all at once; but each product could have been filled out by itself, thus:

Fig. 1.

Fig. 2.

ILLUSTRATION BY SOLIDS.

The number 2167 can be represented by a cubical block, whose volume is found by cubing the number representing the length of one edge, (see Art. 217,) and it is required to find that edge. The first part of the root-viz. 20-is an approximation, exhausting 20 X 20 X 20-8000 units of volume. (See fig. 1.) This leaves 4167 units of volume to be added to the sides of the approximating cube, 8000, to make it equal to 2167, still keeping it a cube.

Therefore, the approximate edge, 20, is, when completed, 20 + 3 23. If 8000 and 4167 had not exhausted the volume, it is plain that the remaining volume should have been applied to the sides of the last formed approximate cube, in the same way as the first remainder was applied; and so with every successive remainder.

Again, if the number had been the pure decimal .012167, it is plain that its three equal component factors would be .23 X .23 X .23, and that the cube of .2 of the root would be in .012, and the cube of the .03 of the root would be in .000167. Hence the decimal should be marked off into groups of three figures from the decimal point toward the right. In a mixed decimal, the integral part must furnish the integral part of the root, and the decimal must furnish the decimal part of the root. Hence such a number should be marked off from the decimal point each way.

Rule. From the place of the decimal point mark off the figures by threes: the last-formed period will have less than three figures when the number of figures is not divisible by 3.

Find the greatest cube in the left-hand period, write its cube root as the first figure of the answer, subtract the cube from the period, and to the remainder annex the second period for a dividend.

Square the root found, annex two ciphers, and multiply the result by 3 for a trial divisor.

Find how many times the trial divisor is contained in the dividend, and write the quotient as the next figure of the root.

To the trial divisor add three times the product of the former part of the root (with a cipher annexed) by the last figure of the root, and the square of the last figure.

Multiply this sum by the last figure of the root, subtract the product, if possible, from the dividend, and to the remainder annex the next period for a new dividend. Proceed as before till all the periods have been used.

NOTE 1.-If the product is greater than the dividend, erase the root figure that produced it, and with a figure of less value recalculate the additions to the trial divisor, till the product is small enough for subtraction.

NOTE 2.-If the trial divisor is not contained in the dividend, annex a cipher to the root, two ciphers to the trial divisor, bring down the next period to the right of the dividend, and proceed as usual.

NOTE 3.-Point off from the right of the root as many figures for decimals as there are decimal periods in the operation.

NOTE 4.-If, in marking off, the last decimal period has less than three figures, fill the period with ciphers.

NOTE 5.-In surds, decimal periods of ciphers may be used to any sufficient degree of approximation.

NOTE 6.-After getting any figure of the root by a trial divisor, we may, for a new dividend, subtract from all the periods that have been used the cube of all the root that has been found. This would be tedious in large operations.

NOTE 7.-The cube root of a perfect cube may be found by resolving it into its prime factors, and finding the product of one of each three equal factors. Thus, 2744 (2 X 2 X 2) X (7 X 7 X 7), and its cube root

=

= 2 X 7 = 14.

NOTE 8.-In finding cube root to a required number of decimal places, the following short methods may be used :

After finding one more than half the required figures, annex one cipher to the last dividend, divide by the trial divisor, take the quotient as a new root figure, multiply the trial divisor by it, subtract the product, annex one cipher to the remainder, and so on. (See Ex. 28.) Or,

After finding one more than half the places, divide the last remainder continually by the last divisor, omitting each time two right-hand figures from the divisor, and one from the remainder. (See Ex. 29.)

18. What is the cube root of 64000000?

19. What is the cube root of 132651000000? 20. What is the cube root of 704969000?

Ans. 400.

Ans. .67. Ans. .012.

21. What is the cube root of .300763?
22. What is the cube root of .000001728?
23. What is the cube root of 1860867?
24. What is the cube root of 389017000?
25. What is the cube root of .000000000000343?

26. Find the value of (343).

Ans. 49.

27. Find the value of

9.16.

Ans. 2.092+.

28. Find the cube root of .01525, correct to 7 decimal places. (Note 8.)

Ans. .2479841+. 29. Find the cube root of .83, correct to 5 decimal places. (Note 8.) Ans. .94103±. 30. Find the cube root of 12, correct to 7 decimal places. 31. Find the value of 144" to 4 places. Ans. 27.47+. 32. Find the value of 5 to 5 places. Ans. 8.5499.

CASE II.

Art. 441. To extract the cube root of a common fraction, or of a mixed number.

PROPOSITION.-The cube root of a fraction is the cube root of its numerator divided by the cube root of its denominator.

DEMONSTRATION.-A fraction is the product of three equal factors, each of which is its cube root. The numerator of the fraction is the product of three equal numerators, and its denominator is the product of three equal denominators. Therefore the cube root of the fraction is a fraction whose numerator is the cube root of its numerator, and whose denominator is the cube root of its denominator.

ILLUSTRATION.-The fraction is the product of three equal fractions, each of which is 2. Hence its numerator 27 is the product of the three numerators, and its denominator 64 is the product of the three denominators. Hence is the cube root of 1.

Rules.-I. For a fraction. Make the cube root of the numerator a new numerator, and the cube root of the denominator a new denominator.

Or,