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FIG. 3.

The volume of the cube marked A, Fig. 1, is 203; the volume of each of the rectangular solids marked B is 20 × 20 × 5, or 202 x 5; the volume of each of the rectangular solids marked C, in Fig. 2, is 20 × 5 × 5, or 20 x 52; and the volume of the small cube marked D is 53. It is evident that if all these solids are put together as represented in Fig. 3, a cube will be formed, each edge of which is 25.

Find the value of each of the following expressions:

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392. The root of a number is either the number itself. or one of its equal factors.

393. Evolution is the process of finding a root of a number, and is the converse of involution.

394. The radical sign is √, or a fractional exponent.

A small figure called the index, written above the radical sign, denotes the root. A fractional exponent is sometimes used. Thus, 16 is equivalent to √/16; 643 to 1/64.

The numerator of the exponent indicates a power, and the denominator a root. Thus, 83 is equivalent to $82.

SQUARE ROOT.

395. The square root of a number is one of its two equal factors. Thus, the square root of 81 is 9, since 9 x9= 81.

PRINCIPLES.-I. The square of a number contains twice as many figures as the number, or twice as many less one. Thus,

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II. If any perfect square be separated into periods of two figures each, beginning with units' place, the number of periods will be equal to the number of figures in the square root of that number.

If the number of figures in the number is odd, the left-hand period will contain only one figure.

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396. To find the square root of a number.

1. Find the square root of 729.

t

t2 + 2 × t× u + u2 = 7,29 ( 20 + 7 = 27

12

= 202 =400

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EXPLANATION.-Since 729 has two periods, its square root has two figures (PRIN. II). Since 29 cannot be a part of the teus, the tens of the root must be found from the first period 7.

The greatest number of tens whose square is contained in 7 is 2. Subtracting the square of 2 tens from 729, the remainder is 329. This remainder is composed of twice the product of the tens of the root by the units, and the square of the units (390). Hence, if 329 be divided by 40, which is twice the tens of the root, the quotient 7 will be either equal to the units' figure of the root, or greater. Subtracting 2x20x7+ 72, or (40+7) × 7, from 329, nothing remains. Hence, 27 is the required root.

In practice, the work may be abridged as follows:

In this example, 40 is the partial or trial divisor, and 47 the complete divisor.

7,29 (27

4

47) 329

329

GEOMETRICAL EXPLANATION OF SQUARE ROOT.

1. Find one side of a square whose area is 729 sq. ft. ?

Let Fig. 1 represent a square whose area is 729 square feet. It is required to find the length of one side of this square.

Since the area of a square is equal to the square of one of its sides, a side may be found by extracting the square root of the area.

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Hence the length of the side

Since 729 consists of two periods, its square root will consist of two figures. The greatest number of tens whose square is contained in 700 is 2. of the square is 20 feet plus the units' figure of the root. Removing the square whose side is 20 ft. and whose area is 400 sq. ft.,

there remains a surface whose area is 329 sq. ft. (Fig. 2). This remainder consists of two equal rectangles, each of which is 20 ft long, and a square whose side is equal to the width of each rectangle. The units' figure of the root is equal to the width of one of these rectangles.

The area of a rectangle is equal to the product of its length and width (215); hence, if the area be divided by the length, the quotient will be the

FIG. 2.

20×7
20

727

400

20×7

width. Now, since the two rectangles contain the greater portion of the 329 sq. ft., 2 x 20 or 40, the length of the two rectangles, may be used as a trial divisor to find the width. Dividing 329 by 40, the quotient is 8. But this quotient is too large for the width of the rectangles, for if 8 ft. is the width, the area of Fig. 2 will be 40 × 8+82 or 384 sq. ft. Taking 7 ft. for the width of the rectangles, the area of Fig. 2 is 40 × 7+72 or 329 sq. ft. Hence, 20+7 or 27 ft. is the length of a side of the square whose area is 729 sq. ft.

If the root contains more than two figures, it may be found by a similar process, as in the following example, where it will be seen that the partial divisor at each step is obtained by doubling that part of the root already found.

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RULE.-1. Separate the given number into periods of two figures each, beginning at the units' place.

2. Find the greatest number whose square is contained in the period on the left; this will be the first figure in the root. Subtract the square of this figure from the period on the left, and to the remainder annex the next period to form a dividend.

3. Divide this dividend, omitting the figure on the right, by double the part of the root already found, and annex the quotient to that part, and also to the divisor; then multiply the divisor thus completed by the figure of the root last obtained, and subtract the product from the dividend.

4. If there are more periods to be brought down, continue the operation in the same manner as before.

If a cipher occur in the root, annex a cipher to the trial divisor, and another period to the dividend, and proceed as before

2. If there is a remainder after the root of the last period is found, annex periods of ciphers and continue the rout to as many decimal places as are required.

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The square root of a fraction may be found by extracting the square root of each of its terms.

If either term of the fraction is an imperfect square, or the number is a mixed number, reduce to the decimal form before extracting the root.

Find the square root of

11. 9.3025. 15. 266256.

19.23. 1369+ √1296.

056

12. 21025. 16. 10795.21. 20. 7958. 24. (2.8)÷√.117649.

216

13. 11881. 17. 196.1369. 21. 17. 14. 104976. 18. .001225. 22. 7.2.

25.

Extending to 4 decimal places, find the

19 32
X

√32 √92

26. √/2. | 27. √/3. | 28. √/8. | 29. √12. | 30. √/26.

31. A square field contains 1016064 square feet. What is the length of each side?

32. A field is 208 rd. long and 13 rd. wide. What is the length of the side of a square containing an equal area?

33. If 251 A. 65 sq. rd. of land are laid out in the form of a square, what will be the length of each of its sides?

34. A circular island contains 21170.25 sq. rd. of land. What is the length of the side of a square field of equal area?

35. If it cost $312 to enclose a field 216 rd. long and 24 rd. wide, what will it cost to enclose a square field of equal area with the same kind of fence?

36. If an army of 55225 men be drawn up in the form of a square, how many men will there be on a side?

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