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Observe, also, that when the figure obtained for the root by dividing, as directed in the fourth part of the rule, is found, on completing the divisor, to be too large, a smaller figure must be substituted in its place, and the divisor completed anew.
There are always as many decimals in the root, as periods of decimals in the power.
We will extract the cube root of 65890311319, in the abridged form; referring, as before, to the particular part of the rule, under which each step of the operation proceeds. First,
65890311319(4039 2dly. Cube of 4, subt'd
1890 3dly. 4X4X3 [div'r]
48 4thly. 48 was not contained in 18.
0000 Lastly. New dividend,
1890311 3dly. 40 X 40 X 3, · 4800 4thly. 4800 in 18903,
3 times. 5thly. Triple product of 40X3,
- 360 Square of 3,
9 Divisor comp’d, 483609 Lastly. 433609 X3, and subtracted,
1450827 New dividend,
439484319 3dlv. 403X403X3,- 487227 4thly. 487227 in 4394
843, 9 times. 5thly Triple product of 403X9,
10881 Square of 9
81 Divisor comp'd, 48831591 Lastly 48831591 X 9, and subtracted,
We will now extract the cube root of 178263.433152, in the abridged form, as in the preceding example; but without reference to the parts of the rule.
1. Extract the cube root of 614125.
To find two MEAN PROPORTIONALS between two given numbers, divide the greater by the less, and extract the cube root of the quotient: then multiply the cube root by the least of the given numbers, and the product will be the least of the mean proportionals; and the least mean proportional multiplied by the same root, will give the greatest mean proportional.
21. What are the two mean proportionals between 6 and 750 ?
22. What are the two mean proportionals between 56 and 12096 ?
To find the side of a cube equal in solidity to any given solid, extract the cube root of the solid contents of the given body, and it will be the required side.
23. There is a stone, of cubic form, containing 21952 solid feet. What is the length of one of its sides?
24. The solid contents of a globe are 15625 cubic inches: required the side of a cube of equal solidity..
25. Required the side of a cubical pile of wood, equal to a pile 28 feet long, 18 st. broad, and 4 ft. high.
All solid bodies are to each other, as the cubes of their diameters, or similar sides.
26. If a ball 6 inches in diameter weighs 32 pounds, what is the diameter of another ball of the same metal, weighing 4 pounds ?
27. If a ball of 4 inches diameter weighs 9 pounds, what is the diameter of a ball weighing 72 pounds ?
28. What must the side of a cubic pile of wood meabure, to contain ğ part as much as another cubic pile, which measures 10 feet on a side ?
29. If 8 cubic piles of wood, each measuring 8 feet on a side, were all put into one cubic pile, what would be the dimensions of one of its sides ?
30. The solid contents of a globe 21 inches in diameter are 4849.0596 solid inches; what is the diameter of a globe, whose solid contents are 11494.0672 inches ?
31. What are the inside dimensions of a cubical bin, that will hold 85 bushels of grain ? (See note, page 27.)
32. What must be the inside dimensions of a cubica]
pin, to hold 450 bushels of potatoes, 2815.489 cubic nches, (heaped measure), making a bushel ?
33. What must be the inside measure of a cubical cistern, to hold 10 hogsheads of water ?
34. What must be the inside measure of a cubica! cistern, that will hold 20 hogsheads of water?
35. What are the inside dimensions of a cubical cis tern, that holds 40 hogsheads of water?
36. Suppose a chest, whose length is 4 feet 7 inches, breadth 2 feet 3 inches, and depth 1 foot 9 inches: what is the side of a cube of equal capacity?
37. Suppose I would make a cubical bin of sufficient capacity to contain 103 bushels; what must be the dimensions of the sides ?
ROOTS OF ALL POWERS.
The roots of many of the higher powers may be extracted by repeated extractions of the square root, or cube root, or both, as the given power may require. Whenever the index of the given power can be resolved into factors, these factors denote the roots, which, being successively extracted, will give the required root.
Thus, the index of the fourth power is 4, the factors of which are 2 X 2; therefore, extract the square root of the fourth power, and then the square root of that square root will be the fourth root. The sixth root is the cube root of the square root, or the square root of the cube root; because 3 X2=6. The eighth roct is the square root of the square root of the square root; because 2 X 2 X 2=3. The ninth root is the cube root of the cube root; because 3x3=9. The tenth root is the fifth root of the square root; because 2 X5=10. The twelfth root is the cube root of the square root of the square root; because 2 X 2 X3=12.
The twenty: seventh root is the cube root of the cube root of the cube root; because 3 X 3 X3=27
The following is a GENERAL RULE for extracting the roots of all powers.
RULE. First- Prepare the given number for extraction, by pointing ofi* from the unit's place, as the required root directs; that is, for the fourth root, into periods of four figures; for the fifth root into periods of five figures, &c.
2dly. - Find the first figure of the root by trial, and subtract its power from the left hand period.
3dly.- To the remainder bring down the first figure in the next period for a dividend.
4thly. -— Involve the root to the next inferior power to that which is given, and multiply it by the number denoting the given power, for a divisor.
5thly- Find how many times the divisor is contained in the dividend, and the quotient will be another figure of the root.
6thly.-Involve the whole root to the given power, and subtract it from the two left hand periods of the given number.
Lastly. Bring down the first figure of the next period to the remainder, for a new dividend, to which find a new divisor, as before. Thus proceed, till the whole root is extracted.
Observe, that when a figure obtained for the root by dividing, is found by involving, to be too great, a less figure must be taken, and the involution performed again. We will extract the fifth root of 36936242722357.
55 = 3125 54 x 5=3125 first divisor. 5686 first dividend.
345025251 614 X 5=-33826005, second divisor. 243371762 2d. dividena.
5175= 36 362427 2357
1 What is the fifth root of 5584059449 ? 2. Find the fifth root of 2196527536224. 3 Extract the fifth root of 16850581551 ? 4. Find the seventh root of 2423162679857794647