748. PROP. II.-The cube of any number must contain three times as many places as the number, or three times as many less one or two places.

This proposition may be shown thus:

1. Observe, 13 = 1, 23 = 8, 33 = 27, 43 = 64, 53 = 125, and 93 = 729; hence the cube of 1 and 2 is expressed each by one figure, the cube of 3 and 4 each by two figures, and any number from 5 to 9 inclusive each by three figures.

2. Observe, also, that for every cipher at the right of a number there must (91) be threc ciphers at the right of its cube; thus, 103 = 1,000, 1003 = 1,000,000. Hence the cube of tens can occupy no place lower than thousands, the cube of hundreds no place lower than millions, and so on with higher orders.

3. From the foregoing we have the following:

(1.) Since the cube of 1 or 2 contains one figure, the cube of 1 or 2 tens must contain four places; of 1 or 2 hundreds, seven places, and so on with higher orders.

(2.) Since the cube of 3 or 4 contains two figures, the cube of 3 or 4 tens must contain five places; of 3 or 4 hundreds, eight places, and so on with higher orders.

(3.) Since the cube of any number from 5 to 9 inclusive contains three places, the cube of any number of tens from 5 to 9 tens inclusive must contain six places; of hundreds, from 5 to 9 hundred inclusive, nine places, and so on with higher orders; hence the truth of the proposition. Hence also the following :

749. I. If any number be separated into periods of three figures each, beginning with the units place, the number of periods will be equal to the number of places in the cube root of the greatest perfect third power which the given number contains.

II. The cube of units contains no order higher than hundreds.

III. The cube of tens contains no order lower than thousands nor higher than hundred thousands, the cube of hundreds no order lower than millions nor higher than hundred millions, and so on with higher orders.

ILLUSTRATION

OF PROCESS.

750. Solution with every Operation Indicated.

Find the cube root of 92345408.

FIRST STEP.

4003 400 x 400 x 400 =

92345408 (400 64000000

(1) Trial divisor 4002 × 3=480000) 28345408 ( 50

(1) Trial divisor 4502 × 3=607500) 1220408 ( 2

EXPLANATION.-1. We place a period over every third figure beginning with the units, and thus find, according to (749), that the root must have three places. Hence the first figure of the root expresses hundreds.

2. We observe that 400 is the greatest number whose cube is contained in the given number. Subtracting 4003 = 64000000 from 92345408, we have 28345408 remaining.

3. We find a trial divisor, according to (746-4), by taking 3 times the square of 400, as shown in (1), second step. Dividing by this divisor, according to (746–5), we find we can add 50 to the root already found. Observe, the root now found is 400+50, and that according to (747),

(400+50)3 = 4003 + 4002 × 50 × 3+502 × 400 × 3+503.

5. We find another trial divisor and proceed in the same manner to find

the next figure of the root, as shown in the third step.

EXPLANATION.-1. Observe, in the first step, we know that the cube of 400 must occupy the seventh and eighth places (749-III). Hence the ciphers are omitted.

2. Observe, also, that no part of the cube of hundreds and tens is found below thousands (749-III). We therefore, in finding the number of tens in the root, disregard, as shown in second step, the right-hand period in the given number, and consider the hundreds and tens in the root as tens and units respectively.

Hence, in general, whatever number of places there are in the root, we disregard, in finding any figure, as many periods at the right of the given number as there are places in the root at the right of the figure we are finding, and consider the part of the root found as tens, and the figure we are finding as units, and proceed accordingly.

From these illustrations we have the following:

752. RULE.-I. Separate the number into periods of three figures each, by placing a point over every third figure, beginning with the units figure.

II. Find the greatest cube in the left-hand period, and place its root on the right. Subtract this cube from the period and annex to the remainder the next period for a dividend.

III. Divide this dividend by the trial divisor, which is 3 times the square of the root already found, con

sidered as tens; the quotient is the next figure of the root.

IV. Subtract from the dividend 3 times the square of the root before found, considered as tens, multiplied by the figure last found, plus 3 times the square of the figure last found, multiplied by the root before found, plus the cube of the figure last found, and to the remainder annex the next period, if any, for a new dividend.

V. If there are more figures in the root, find in the same manner trial divisors and proceed as before.

In applying this rule be particular to observe:

1. In dividing by the Trial Divisor the quotient may be larger than the required figure in the root, on account of the addition to be made, as shown in (746–6) second step. In such case try a figure 1 less than the quotient found.

2. When there is a remainder after the last period has been used, annex periods of ciphers and continue the root to as many decimal places as may be required.

3. We separate a number into periods of three figures by beginning at the units place and proceeding to the left if the number is an integer, and to the right if a decimal, and to the right and left if both.

4. Mixed numbers and fractions are reduced to decimals before extracting the root. But in case the numerator and the denominator are perfect third powers, or the denominator alone, the root may be more readily found by extracting the root of each term separately

17. Find, to two decimal places, the cube root of 11. Of 36.

Of 84. Of 235.

Of

Of 5. Of 75.4. Of 6.7.

9. 344. 10. 5832

13. 24137569.

14. 47045881.

12. 438976.

18. Find to three decimal places the cube root of 3. Of .5. Of .04. Of .009. Of 2.06.

19. Find the sixth root of 4096.

Of 7.

Observe, the sixth root may be found by extracting first the square root, then the cube root of the result.

For example, 4096 = 64; hence, 4096-64 × 64. Now, if we extract the cube root of 64 we will have one of the three equal factors of 64, and hence one of the six equal factors or sixth root of 4096.

Thus, 644; hence, 64

its square root that 4096

=

4x4x4. But we found by extracting 64 × 64, and now by extracting the cube root

that 64=4×4×4 ; consequently we know that 4096=(4 × 4 × 4) × (4 × 4 × 4). Hence 4 is the required sixth root of 4096.

In this manner, it is evident, we can find any root whose index contains no other factor than 2 or 3.

20. Find the sixth root of 2565726409.

21. Find the eighth root of 43046721.
22. What is the fourth root of 34012224?
23. What is the ninth root of 134217728?

24. A pond contains 84604519 cubic feet of water; what must be the length of the side of a cubical reservoir which will exactly contain the same quantity.

25. What is the length of the inner edge of a cubical cistern that contains 2079 gal. of water?

26. How many square feet in the surface of a cube whose volume is 16777216 cubic inches.

27. A pile of cord wood is 256 ft. long, 8 ft. high, and 16 ft. wide; what would be the length of the side of a cubical pile containing the same quantity of wood?

28. What is the length of the inner edge of a cubical bin that contains 3550 bushels ?

29. What are the dimensions of a cube whose volume is equal to 82881856 cubic feet?

30. What is the length in feet of the side of a cubical reservoir which contains 1221187.5 pounds avoirdupois, pure

water?