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EXTRACTION OF THE SQUARE ROOT.

350. To extract the square root, is to resolve a given number into two equal factors; or, to find a number which being multiplied into itself, will produce the given number. (Art. 344. Obs.)

Ex. 1. What is the side of a square room which contains 16 square yards?

*Solution. Let the room be represented by the adjoining figure. It is divided into 16 equal squares, which we will call square yards.

4 yards.

4 yards.

Since the room is square, the question is simply this: What is the square root of 16? Now if we resolve 16 into two equal factors, each of those factors will be the square root of 16. But 16=4 X4. The square root of 16, therefore, is 4.

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4X4 16 yards.

2. What is the length of one side of a square room which contains 576 square feet?

Operation. Since we may not see what the 576(24 root of 576 is at once, as in the last

4

44)176

176

example, we will separate it into periods of two figures each, by putting a point over the 5, and also over the 6; that is, over the units' figure and over the hundreds. This shows us that the root is to have two figures; (Art. 342. Obs 2 ;) and thus enables us to find the root of part of the number at a time. Now the greatest square of 5, the left hand period, is 4, the root of which is 2. We place the 2 on the right hand of the number for the first part of the root; then subtract its square from 5, the period under consideration, and to the right of the remainder bring down 76, the next period, for a dividend. To find the next figure in the root, we double the 2, the part of the root already found, and placing it on the left QUEST.-350. What is it to extract the square root of a number?

of the dividend for a partial divisor, we find how many times it is contained in the dividend, omitting the right hand figure. Now 4 is contained in 17, 4 times. Placing the 4 on the right of the root, also on the right of the partial divisor, we multiply 44, the divisor thus completed, by 4, the last figure in the root, and subtracting the product 176 from the dividend, find there is no remainder. The answer therefore is 24.

Note. Since the root is to contain two figures, the 2 stands in tens' place; hence the first part of the root found is properly 20; which being doubled, gives 40 for the divisor. For convenience we omit the cipher on the right; and to compensate for this, we omit the right hand figure of the dividend. This is the same as dividing both the divisor and dividend by 10, and therefore does not alter the quotient. (Art. 88.) PROOF. 24-2 tens, or 20+4 units.

[blocks in formation]

(24)2=576 =400+160 16. (Art. 355.)

ILLUSTRATION BY GEOMETRICAL FIGURE.

[blocks in formation]

20ft.

F

4ft.

20ft.

room in the last example; then the square DEFG will be the greatest E square of the left hand. period, the root of which is 20 ft., and 20×20= 400, the number of feet in its area. (Art. 163.) But this square 400 ft. taken from 576 ft. leaves a remainder of 176 ft. Now it is plain if this remaining space is all added to one side of this square, its sides would become unequal; consequently it would cease to be a square. (Art. 153. Obs. 1.) But if it is equally enlarged on two sides it will obviously continue to be a

D

20ft.

G

C

QUEST.-Note. What place does the first figure of the root occupy in the example above? Why is the right hand figure of the dividend omitted?

square.

For this reason the root is doubled for a divisor in the operation. The parallelograms AEFH and GFIC will therefore represent the additions made to the two sides, each of which is 4 ft. wide; consequently the area of each is 20×480 ft., and the area of both is 40 x 4 160 ft.

But having made these additions to two sides of the square, there is a vacancy at the corner. The square BIFH represents this vacancy, the side of which is 4ft., or the same as the width of the additions; and its area is 4×4=16 ft. For convenience of finding the area of this vacancy, it is customary in the operation to place the last figure of the root on the right of the divisor, and thus it is multiplied into itself. The figure is now a perfect square, the length of whose side is 20+4=-24 ft.

351. From these principles and illustrations we derive the following general

RULE FOR EXTRACTING THE SQUARE ROOT.

I. Separate the given number into periods of two figures each, by placing a point over the units' figure, another over the hundreds, and so on over each alternate figure.

II. Find the greatest square number in the first or left hand period, and place its root on the right of the number for the first figure in the root. Subtract the square of this figure of the root from the period under consideration; and to the right of the remainder bring down the next period for a dividend.

III. Double the root just found and place it on the left of the dividend for a partial divisor, find how many times it is contained in the dividend, omitting its right hand figure; place the quotient on the right of the root, also on the right of the partial divisor; multiply the divisor thus completed by the last figure of the root; subtract the product from the dividend, and to the remainder bring down the next period for a new dividend, as before.

IV. Double the root already found for a new partial di

QUEST.-351. What is the first step in extracting the square root? The second? Third? Fourth?

visor, divide, &c. as before, and thus continue the operation till the root of all the periods is extracted.

PROOF.-Multiply the root into itself; and if the product is equal to the given number, the work is right. (Art. 344.)

OBS. The reason for the several steps in the rule may easily be inferred from the preceding illustrations. The following is a summary of them:

1. Separating the given number into periods of two figures each shows how many figures the root is to contain, (Art. 348. Obs. 3,) and thus enables us to find part of the root at a time.

2. The square of the first figure of the root is the number of feet, yards, &c. disposed of by the first figure of the root; it is subtracted from the period to find how many feet, yards, &c., remain to be

added.

3. The root thus found is doubled for a partial divisor, because the addition must be made on two sides of the square already found, or it will cease to be a square.

4. In dividing, the right hand figure of the dividend is omitted, because the cipher on the right of the divisor is omitted; otherwise the quotient would be 10 times too large for the next figure in the root.

5. The last figure of the root is placed on the right of the divisor for convenience of multiplying. The divisor is then multiplied by the last figure of the root to find the area of several additions thus made. 3. What is the square root of 625 ? 4. What is the square root of 900 ? 5. What is the square root of 1225? 6. What is the square root of 1764 ? 7. What is the square root of 2916 ? 8. What is the square root of 4761 ? 9. What is the square root of 8649 ? 10. What is the square root of 12321? 11. What is the square root of 53824? 12. What is the square root of 531441 ?

QUEST.-How is the square root proved? Obs. Why do we separate the given number into periods of two figures each? Why subtract the square of the first figure in the root from the first period? Why double the root thus found for a divisor? Why omit the right hand figure of the dividend? Why place the last figure of the root on the right of the divisor? Why multiply the divisor by the last figure in the root? 352. When there are decimals in the given number, how are they pointed off? When there is a remainder, how proceed?

352. If there are decimals in the given sum, they must be separated into periods like whole numbers, by placing a point over units, then over hundredths, and so on, over every alternate figure towards the right.

If there is a remainder after all the periods are brought down, the operation may be continued by annexing periods of ciphers.

OBS. 1. There will always be as many decimal figures in the root, as there are periods of decimals in the given number.

2. The square root of a common fraction is found by extracting the root of the numerator and denominator.

3. A mixed number should be reduced to an improper fraction. When either the numerator or denominator of a common fraction is not a square number, the fraction may be reduced to a decimal and the approximate root be found as above.

13. What is the square root of 6.25? Ans. 2.5.
14. What is the square root of 1.96?

15. What is the square root of 29.16?
16. What is the square root of 234.09?
17. What is the square root of .1225?
18. What is the square root of .776161 ?
19. What is the square root of 2?
20. What is the square root of 17?

21. What is the square root of 175 ?
22. What is the square root of 116964?
23. What is the square root of 10316944?

36

24. What is the square root of 28?

49

25. What is the square root of 44 ? 26. What is the square root of 61? 27. What is the square root of 52%?

QUEST.-Obs. How do you determine how many decimal figures there should be in the root? How is the square root of a common fraction found? Of a mixed number?

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