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46656 contents of the whole monument.

1. Required the cube root of 77303776.

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2. What is the cube root of 34965783 ? 3. What is the cube root of 436036824287? 4. What is the cube root of 84.604519 ? 5. Required the cube root of 54439939. 6. Extract the cube root of 60236288. 7. Extract the cube root of 109215352. 8. What is the cube root of 116.930169 ? 9. What is the cube root of .726572699 ? 10. Required the cube root of 2. 11. Find the cube root of 11.

Ans. 327. Ans. 7583. Ans. 4.39.

Ans. 379.

Ans. 392.

Ans. 478.

Ans. 4.89. Ans. .899. Ans. 1.2599+.

Ans. 2.2239+.

12. What is the cube root of 122615327232 ? Ans. 4968.

13. What is the cube root of 125? 14. What is the cube root of 1831? 15. What is the cube root of 12?

4913

729

16. What is the cube root of $9319

68921

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Ans. .

Ans. 11.

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To find the cube root of any number mentally, less than 1,000,000, when the number has an exact root.

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RULE. As there will be two figures in the root, the first may easily be found mentally, or by the table of powers; and if the unit figure of the power be 1, the unit figure in the root will be 1; and if it be 8, the root will be 2; and if 7 it will be 3; and if the unit of the power be 6, the unit of the root will be 6; and if 5, it will be 5; if 3, it will be 7; if 2, it will be 8; and if the unit of the power be 9, the unit of the root will be 9. This will appear evident by inspecting the table of powers.

Ans. 46. period, we find

17. What is the cube root of 97336 ? Explanation. By examining the left-hand the root of 97 is 4, and the cube of 4 is 64. The root cannot be 5, because the cube of 5 is 125. The unit of the power is 6; therefore, by the above rule, the unit figure in the root is 6. The answer, therefore, is 46.

18. What is the cube root of 132651 ? 19. What is the cube root of 148877 ? 20. What is the cube root of 175616? 21. What is the cube root of 185193? 22. What is the cube root of 238328? 23. What is the cube root of 262144? 24. What is the cube root of 389017? 25. What is the cube root of 405224? 26. What is the cube root of 531441 ? 27. What is the cube root of 24389 ? 28. What is the cube root of 42875?

Ans. 51.

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SECOND METHOD OF EXTRACTING THE CUBE ROOT.

The following rule for the extraction of the root of the third power, though it is essentially the same with the former, may yet serve to make the reasons for the several steps of the operation more intelligible to the learner.

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RULE. Separate the number whose root is to be found into periods, as under the former rule, and find by trial the greatest root in the lefthand period, and put it in the place of the quotient.

Subtract the third power of this root from the period to which it belongs, and to the remainder bring down the next period for a dividend.

Then, to find a divisor, annex a cipher to the root already found, and multiply twice the number thus formed by the number itself, and to the product add the second power of this number.*

This is the same as multiplying the square of the radical figure by 300, as in the former rule.

Ascertain how many times this divisor is contained in the dividend, and write the result in the quotient.*

Then, to find the subtrahend, multiply this divisor by its quotient, and write the product under the dividend. To this add three times the preceding radical figure with a cipher annexed,† multiplied by the second power of the figure last obtained, and also the third power of this last figure. Subtract the sum of their several products from the dividend above them, and to the remainder bring down the next period for a new dividend. With the parts of the root already found proceed to find a divisor and subtract as above, and so on, till the successive figures of the root are all obtained.

The rationale of the above rule may be made to appear by the solution of the following question.

Let it be required to find the cube root of 17576.

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We now raise the quantity 20+ 6 to the third power.

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*This quotient figure must sometimes be less than the one indicated by the divisor, and in extreme cases the divisor may give a quotient too large by several units. The quotient required can, of course, never exceed 9.

This accounts for multiplying by 30, in the foregoing rule, which is a factor in the triple quotient in finding the subtrahend.

Now, by observing this operation, and remarking what would be lost in the course of it by omitting the second figure of the root, 6, taking 20 instead of 26, we see that when we have found the 20 the next inquiry is, what number must be added to 20, so that, if we multiply it into itself once and into 20 twice, and the sum of these products, together with the second power of twenty, by twenty plus this number, the result will be 17576, or 8000 +9576. Then, in order to obtain this number, or an approximation to it, we take twice the part of the root found, 20, and multiply the result, 40, by 20, as we should do in raising it to the third power, and make this, which is 800, a part of the divisor, and the product of which by 6 was lost in the operation for want of the 6 added to 20. But this is not all the loss. There is also the second power of 20 by 6, and therefore 400 to be added to the 800 for a divisor. There still remains the further loss of the third power of 6 (216), and also of 6 times 240 and 20 times 36; but these we neglect in the formation of the divisor. The divisor is contained in the dividend 7 times; but, making the allowance of a unit for the neglect of the numbers above named, we take 6 for the quotient figure, and proceed to find the subtrahend, which, according to the rule and the foregoing operation of raising 20 +6 to the third power, must be 1200 × 6+1440 +720+216= 17576.

APPLICATION OF THE CUBE ROOT.

PRINCIPLES ASSUMED.

Spheres are to each other as the cubes of their diameter. Cubes, and all solids whose corresponding parts are similar. and proportional to each other, are to each other as the cubes of their diameters, or of their homologous sides.

29. If a ball, 3 inches in diameter, weigh 4 pounds, what will be the weight of a ball that is 6 inches in diameter?

Ans. 32lbs. 30. If a globe of gold, one inch in diameter, be worth $120, what is the value of a globe 3 inches in diameter ?

Ans. $5145.

31. If the weight of a well-proportioned man, 5 feet 10 inches in height, be 180 pounds, what must have been the weight of Goliath of Gath, who was 10 feet 4 inches in height? Ans. 1015.1+lbs. 32. If a bell, 4 inches in height, 3 inches in width, and of

an inch in thickness, weigh 2 pounds, what should be the dimensions of a bell that would weigh 2000 pounds?

Ans. 3ft. 4in. high, 2ft. 6in. wide, and 24in. thick. 33. Having a small stack of hay, 5 feet in height, weighing lcwt., I wish to know the weight of a similar stack that is 20 feet in height. Ans. 64cwt.

34. If a man dig a small square cellar, which will measure 6 feet each way, in one day, how long would it take him to dig a similar one that measured 10 feet each way?

Ans. 4.629days.

35. If an ox, whose girth is 6 feet, weighs 600lbs., what is the weight of an ox whose girth is 8 feet? Ans. 1422.2+lbs. 36. Four women own a ball of butter, 5 inches in diameter. It is agreed that each shall take her share separately from the surface of the ball. How many inches of its diameter shall

each take?

Ans. First, .45+ inches; second, .57+ inches; third, .82 inches; fourth, 3.149+- inches.

37. John Jones has a stack of hay in the form of a pyramid. It is 16 feet in height, and 12 feet wide at its base. It contains 5 tons of hay, worth $17.50 per ton. Mr. Jones has sold this hay to Messrs. Pierce, Rowe, Wells, and Northend. As the upper part of the stack has been injured, it is agreed that Mr. Pierce, who takes the upper part, shall have 10 per cent. more of the hay than Mr. Rowe; and Mr. Rowe, who takes his share next, shall have 8 per cent. more than Mr. Wells; and Mr. Northend, who has the bottom of the stack, that has been much injured, shall have 10 per cent. more than Mr. Wells. Required the quantity of hay, and how many feet of the height of the stack, beginning at the top, each receives.

Ans. Pierce receives 275cwt. and 10.366+ feet in height; Rowe, 2419cwt. and 2.493 feet; Wells, 22344cwt. and 1.666 feet; Northend, 253cwt. and 1.474 feet.

A GENERAL RULE FOR EXTRACTING THE ROOTS OF ALL POWERS.

RULE.-1. Prepare the given number for extraction, by pointing off from the unit's place, as the required root directs.

2. Find the first figure of the root by trial, or by inspection, in the table of powers, and subtract its power from the left-hand period.

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