CUBE ROOT.

680. Oral Exercises.

a. What is one of the three equal factors of 8? of 27? of 64? of 125? of 216? of 1000? of 1728 ?

b. What is the cube root of 0.001 ? of 0.008? of 0.027? of 0.216? of 1.728?

To find the Number of Terms in the Cube Root of a given

Number.

681. By cubing 1 and 9, 10 and 99, 100 and 999, etc., we obtain the following results:

These results show that when a number has one term, its cube is expressed by one, two, or three figures; when a number has two terms, its cube is expressed by four, five, or six figures; when a number has three terms, its cube is expressed by seven, eight, or nine figures, and so on. Hence,

682. If a numerical expression be separated into periods of three figures each, beginning with the units' figure, the number of periods will be the same as the number of terms in the cube root.

To find the Parts of a Third Power.

683. To find what parts a third power is made up of, we may take a number consisting of tens and units, 36, for example, and raise it to the third power.

The foregoing written work shows the partial products obtained by multiplying each term of the square of 36 by each term of 36.

From this we see that the third power of a number consisting of tens and units is made up of four parts:

(1.) The cube of the tens.

(2.) Three times the product of the square of the tens by the units. (3.) Three times the product of the tens by the square of the units. (4.) The cube of the units.

These parts may be expressed by the formula,

Tens3 + 3 (tens2 × units) + 3 (tens × units2) + units.3

the square of the tens by the units, plus three times the product of the tens by the square of the units, plus the cube of the units.

As the first part of the power, the cube of tens, is thousands, we find the greatest cube contained in 262 (thousands), which is 216 (thousands), and write its cube root 6 (tens) as the first term, or tens, of the root.

Taking the cube of 6 (tens), 216 (thousands), out of 262 (thousands), there remain 46 (thousands).

Now, as the second part of the power, three times the product of the square of the tens by the units, is hundreds, we unite the 1 hundred of the given number with the 46 thousands remaining, and have 461 hundreds.

*For another method of finding cube roots, see Appendix, page 313.

This 461 hundreds has in it a product of which three times the square of the tens of the root is one factor, and the units of the root is the other factor. Dividing 461 (hundreds) by three times the square of the tens, or 108 (hundreds), we find 4 (units) to be the other factor, which we write as the second term, or units of the root.

Multiplying 108 (hundreds) by 4 (units) and taking the product 432 (hundreds) out of 461 (hundreds), we have 29 (hundreds) left.

Now, as the third part of the power, three times the product of the tens by the square of the units, is tens, we unite 4 (tens) of the given number with the 29 (hundreds) remaining, and have 294 (tens).

We take out of this number three times the product of the tens by the square of the units, 288 (tens), and have 6 (tens) left.

Now, as the fourth part of the power, the cube of the units, is units, we unite the 4 units of the given number with the 6 tens remaining, and have 64.units.

We take the cube of the 4 units out of this number, and nothing remains. Ans. 64.

126/732.947/167 (50.23 number 126732.947167,

we proceed as in Illustrative Example I. Having found this part of the root, we consider it as tens in reference to the next term, take three times its square for a new divisor, form a new dividend, and proceed as before.

So, whatever the number of terms in the root, having found a part of them, we consider that part to be the tens, take three times its square for a new divisor, and proceed as before.

686. Rule.

To extract the cube root of a number:

1. Beginning with the units' figure, point off the expression into periods of three figures each.

2. Find the greatest cube in the number expressed by the left-hand period, and write its cube root as the first term of the root.

3. Subtract this cube from the part of the number used, and with the remainder unite the next term of the given number for a dividend.

4. Take three times the square of the part of the root already found for a divisor, and by this divide the dividend, writing the quotient as the next term of the root.*

5. Multiply the divisor by this term, and subtract the product from the dividend. With the remainder unite the next term of the given number.

6. Subtract from the number thus formed three times the product of the first term of the root by the square of the second. With this remainder unite the next term of the given number.

7. Subtract from the number thus formed the cube of the second term of the root.

8. If there are more terms of the root to be found, unite with the remainder the next term of the given number, take for a divisor three times the square of the part of the root now found, and proceed as before.

NOTE I. When, as in the work of Illustrative Example II., the divisor is not contained in the dividend, place a zero as the next figure in the root; also place two zeros at the right of the divisor, and for a neu dividend unite the next three terms of the given number with the previous dividend.

*If this quotient should prove to be too large, make it less and repeat the work.

NOTE II. The cube root of a common fraction whose numerator and denominator are both perfect cubes may be found by taking the cube root of the numerator and of the denominator.

But when the numerator or the denominator of the fraction is not a perfect cube, change the fraction to a decimal and then extract its cube root.

NOTE III. When there is a remainder after all the terms of the given number have been used, annex zeros, and continue the operation as far as desired.

687. Examples for the Slate.

Roots of number: not perfect cubes may be found to the fourth term.

74. What is the length of one edge of a cube which contains 36594368 cubic inches?

75. What is the length of a cubical pile of wood which contains 2 cords?

76. What is the depth of a cubical cistern which contains 300 gallons?

77. What is the depth of a cubical bin which will hold 65 bushels of grain?

78. What will be the cost of lead, at 12 cents per lb., 1 lbs. to the square foot, to line a cubical box containing 15ğ cubic feet?

689. Exercises upon Drill Table No. 2 (page 60).

247. Find the sq. root of the numbers given in column B (25 examples). 248. Find the cu. root of the numbers given in column E (25 examples).