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More Exercises for the Slate.

16. What is the square root of 65536?

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16. What is the square root of 470596? A. 686.
19. What is the square root of 1048576? A. 1024.
20. What is the square root of 2125764? A. 1453.
21. What is the square root of 6718454? A. 2592.
22. What is the square root of 23059204? A. 4802.
23. What is the square root of 4294967296? A. 65536.
24. What is the square root of 40?

In this example, we have a remainder, after obtaining one figure in the root In such cases, we may continue the operation to decimals, by annexing two ciphers for a new period, and thus continue the operation to any assignable degree of exactness. But since the last figure, in every dividend thus formed, will always be a cipher, and as there is no figure under 10 whose square number ends in a cipher, there will, of course, be a remainder; consequently, the pupil need not expect, should he continue the operation to any extent, ever to obtain an exact root. This, however, is by no means necessary; for annexing only one or two periods of ciphers will obtain a root sufficiently exact for almost any purpose. A. 6,3245 +

25. What is the square root of 30? A. 5,4772.

26. What is the square root of ·14 A. 13.

Or, we may reduce the given fraction to its lowest terms before the root is extracted.

Thus, † = 3, Ans., as before.

9. What is the square root of 45? A. 15.

2048

28. What is the square root of ? A. §.

29. What is the square root of 172T? A. TOT.

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If the fraction be a surd, the easiest method of proceeding will be to reduce to a decimal first, and extract its root afterwards.

30. What is the square root of ? A.,9128+

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32. What is the square root of 9? A.,83205.

33. What is the square root of 4201?

In this example, it will be best to reduce the mixed number to an improper fraction, before extracting its root, after which it may be converted into a mixed number again. A. 201.

34. What is the square root of 912? A. 30.

35. A general has an army of 5625 men; how many must he place in rank and file, to form them into a square? ✅✔✅5625=75, Âns.

36. A square pavement contains 24336 square stones of equal size; how many are contained in one of its sides? A. 156.

37. In a circle, whose area, or superficial contents, is 4096 feet, I demand what will be the length of one side of a square containing the same number of feet? A. 64 feet.

38. A gentleman has two valuable building spots, one containing 40 square rods, and the other 60, for which his neighbor offers him a square field, containing 4 times as many square rods as the building spots; how many rods in length must each side of this field measure? 40+60 × 4 = 20, Ans.

39. How many trees in each row of a square orchard, containing 14400 trees? A. 120 trees.

40. A certain square garden spot measures 4 rods on each side; what will be the length of one side of a garden containing 4 times as many square rods? A. 8 rods. 41. If one side of a square piece of land measure 5 rods, what will the side of one measure, which is four times as large? 16 times as large? 36 times as large? A. 10. 20. 30.

42. A man is desirous of forming a tract of land, containing 140 acres, 2 foods and 20 rods, into a square; what will be the length of each side?

A. 150 rods.

43. The distance from Providence to Norwich, Conn., is computed to be 45 miles; now, allowing the road to be 4 rods wide, what will be the length of one side of a square lot of land, the square rods of which shall be equal to the equare rous contained in said road? A. 240 rods.

EXTRACTION OF THE CUBE ROOT.

T LXXXVII. Q. Involution, (T LXXXIV.,) you doubtless recol lect, is the raising of powers; can you tell me what is the 3d power of 3, and what the power is called?

A. 27, called a cube.

Q. Evolution ( LXXXVII.) was defined to be the extracting the 1st power ar roots of higher powers; can you tell me, then, what is the cube root of 27? A. 3.

Q. Why?

A. Because 3x3 × 3, or, expressed thus, 33=27.

Q. What, then, is it to extract the cube root of any number? A. It is only to find that number, which, being multiplied into itself three times, will produce the given number.

Q. We have seen, ( LXXX.,) hat, to find the contents of solid bodies, such as wood, for instance, we mutiply the length, breadth and depth to gether. These dimensions are called cubic, because, by being thus multiplied, they do in fact contain so many solid feet,ches, &c., as are expressed by thei product but what do you suppose the shape of a solid body is, which is an

exact cube?

A. It must have six equal sides, and each side must be an exact square. See block A, which accompanies this

work.

Q. Now, since the length, breadth and thickness of any regular cube are exactly alike, as, for instance, a cubical block, which contains 27 cubic feet, can you inform me what is the length of one side of this block, and what the length may be called?

A. Each side is 3 feet, and may be called the cube root of 27.

Q. Why? A. Because 33 =27.

Q. What is the length of each side of a cubical block containing 64 cubic inches? A. 4 inches.

Q. Why? A. Because 4 X 4 X 4, or 43 64 cubic inches.
Q. What is the cube root of 64, then? A. 4.
Q. Why? A. Because 43 = 64.

Q. What is the length of each side of a cubical block containing 1000 cubic feet? A. 10.

Q. Why? A. Because 103 1000.

1. In a square box which will contain 1000 marbles, how many will it take to reach across the bottom of the box, in a straight row? A. 10.

2. What is the difference between the cube root of 27 and the cube of 3?

A. 24.

3. What is the difference between 3/8 and 23?

A. 6.
A. 0.

4. What is the difference between 3/1 and 13? 5. What is the difference between the cube root of 27 and the square root of 9? A. 0.

6. What is the difference between 3/8 and 4? A 0.

Operation by Slate Illustrated.

7. A man, having a cubical block containing 13824 cubic feet, wishes to know the length of each side, without measuring it; what is the length of each side of said block?

Should we attempt to illustrate the reason of the rule for extracting the cube root, by exhibiting the picture of the cube and its various parts on paper, it would tend rather to confuse than illustrate the subject. The best method of doing it is, by making several small blocks, which may be supposed to contain a certain proportional number of feet, inches, &c., corresponding with the operation of the rule. They may be made in a few minutes, from a small strip of a pine board, with a common penknife, at the longest, in less time than the teacher can make the pupil comprehend the reason, from merely seeing the picture on paper. In demonstrating the rule in this way, it will be an amusing and instructive exercise, to both teacher and pupil, and may be comprehended by any pupil, however young, who is so fortunate as to have progressed as far as this rule. It will give him distinct ideas respecting the different dimensions of square and cubic measures, and indelibly fix on his mind the reason of the rule, consequently the rule itself. But, for the convenience of teachers, blocks, lustrative of the operation of the foregoing example, will accompany this

work.

The following are the supposed proportional dimensions of the several blocks used in the demonstration of the above example, which, when put together, ought to make an exact cube, containing 13824 cubic feet:

One block, 20 feet long, 20 feet wide, and 20 feet thick; this we will call A Three small blocks, each 20 feet long, 20 feet wide, and 4 feet thick; each of these we will call B.

Three smaller blocks, each 20 feet long, 4 feet wide, and 4 feet thick; each of those we will call C.

One block, and the smallest, 4 feet long, 4 feet wide, and 4 feet thick; this we will call D.

We are now prepared to solve the preceding example.

In this example, you recollect, we were to find the length of one side of the sube, containing 13824 cubic feet.

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Or,

64

5824 deducted.

0000

In this example, we know that one side cannot be 30 feet, for 30327000 solid feet, being more than 13824, the given sum; therefore, we will take 20 for the length of one side of the cube.

Then, 20 x 20 x 20 = 8000 solid feet, which we must, of course, deduct from 13824, leaving 5824. (See Operation 1st.) These 8000 solid feet, the pupil will perceive, are the solid contents of the cubical block marked A. This cor

The same operation, by neglecting the ciphers, responds with the operation; may be performed thus:

OPERATION 2d.

13824 (20+4, or 24, root,

8

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for we write 20 feet, the length of the cube A, at the right of 13824, in the form of a quocient; and its square, 8000,

Ans. under 13824; from which subtracting 8000, leaves 5824, as before.

As we have 5824 cubic feet remaining, we find the sides of the cube A are not so long as they ought to be; consequently we must enlargo A; but in doing this, we must enlarge the three sides of A, in order that we may preserve the

400,

cubical form of the block. We will now place the three blocks, each of which is marked B, on the three sides of A. Each of these blocks, in order to fit, must be as long and as wide as A; and, by examining them, you will see that this is the case; that is, 20 feet long and 20 feet wide; then 20 x 20 the square contents in one B; and 3 × 400 = 1200, square contents in 3 Bs; then it is plain, that 5824 solid contents, divided by 1200, the square contents, will give the thickness of each block. But an easier method is, to square the 2, (tens,) in the root 20, making 4, and multiply the product, 4, by 300, making 1200, a divisor, the same as before.

We do the same in the operation (which see); that is, we multiply the square of the quotient figure, 2, by 300, thus, 2 x 24 x 300 1200; then the divisor, 1200 (the square contents) is contained in 5824 (solid contents) 4. times; that is, 4 feet is the thickness of each block marked B. This quotient figure, 4, we place at the right of 5824, and then, 1200 square feet X 4 feet, the thickness, 4800 solid feet.

If we now examine the block, thus increased by the addition of the 3 Ba, we shall see that there are yet three corners not filled up: these are represented by the three blocks, each marked C, and each of which, you will per ceive, is as long as either of the Bs, that is, 20 feet, being the length of A, which is the 20 in the quctient. Their thickness and breadth are the same as the thickness of the Bs, which we found, by dividing, to be 4 feet, the last quotient figure. Now, to get the solid contents of each of these Cs, we multiply their thickness (4 feet) by their bread: (4 feet), 16 square feet; that is, the square of the last quotient figure, 4, 16; these 16 square contents must be

multiplied by the length of each, (20 feet,) or, as there are 3, by 3 x 20 = 60, or, which is easier in practice, we may multiply the 2 (tens), in the root, 20, by 30, making 60, and this product by 42 16, the square contents = 960 solid

feet.

We do the same in the operation, by multiplying the 2 in 20 by 30 = 60 × 4 X 4 = 960 solid feet, as before; this 960 we write under the 4800, for we must add the several products together by and by, to know if our cube will contain all the required feet.

By turning over the block, with all the additions of the blocks marked B and C, which are now made to A, we shall spy a little square space, which prevents the figure from becoming a complete cube. The little block for this corner is marked D, which the pupil wall find, by fitting it in, to exactly fill up this space. This block D is exactly square, and its length, breadth, and thickness are alike, and, of course, equal to the thickness and width of the Cs, that is, 4 feet, the last quotient figure; hence, 4 ft. X 4 ft. X 4 ft. 64 solid feet in the block D; or, in other words, the cube of 4, (the quotient figure,) which is the same as 4364, as in the operation. We now write the 64 under the 960, that this may be reckoned in with the other additions.

We next proceed to add the solid contents of the Bs, Cs, and D together, thus, 4500+960 + 645824, precisely the number of solid feet which we had remaining after we deducted 8000 feet, the solid contents of the cube A.

If, in the operation, we subtract the amount, 5824, from the remainder of dividend, 5824, we shall see that our additions have taken all that remained after the first cube was deducted, there being no remainder.

The last little block, when fitted in, as you saw, rendered the cube complete, each side of which we have now found to be 204-4=24 feet long which is the cube root of 13824 (solid feet); but let us see if our cube contain the required number of solid feet.

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In Operation 2d, we see, by neglecting the ciphers at the right of 8, the 8 is still 8000, by its standing under 3 (thousands); hence, we may point off three figures by placing a dot over the units, and another over the thousands, and

so on.

From the preceding example and illustrations we derive the following

RULE.

Divide the given number into periods of three figures each, by placing a point over the unit figure, and every third figure from the place of units to the left, in whole numbers, and to the right in decimals.

Find the greatest cube in the left-hand period, and place its root in the quotient.

Subtract the cube thus found from the said period, and to the remainder bring down the next period, and call this the dividend.

Multiply the square of the quotient by 300, calling it the divisor.

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