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over it, and the whole placed before the given number; or by a fractional index or exponent, placed over, and a little to the right of the given number.

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271. Evolution teaches to find the roots of any given number.

Thus to extract the square root of a number, is to find such a number which being multiplied once into itself, produces the given number: to extract the cube root, is to find a number which being multiplied into itself, and that product into the same number, produces the given number; and so for other roots.

EXTRACTION OF THE SQUARE ROOT.

272. The following table contains the first nine whole numbers which are exact squares, with the square root of each placed under its respective number.

SQUARES... . . . . . l. 4. 9. 16. 25. 36, 49. 64, 81. SQUARE ROOTS. l. 2. 3. 4. 5. 6. 7. S. 9.

From this table the root of any exact square, being a single figure, may be obtained by inspection, as is plain.

273. To extract the square root, when it consists of two or
more figures, from any number.

Rule I. Make a point over the units place of the given number, another over the hundreds, and so on, putting a point over every second figure; whereby the given number will be divided into periods of two figures each, except the left hand period, which will be either one or two, according as the number of figures in the whole is odd or even. II. Find in the table the greatest square number not greater than the left hand period, set it under that period, and its root in the quotient. III. Subtract the said square from the figures above it, and to the remainder bring down the next period for a dividend. IV. Double the root, (or quotient figure,) and place it for a divisor on the left of the dividend. V. Find how often the divisor is contained in the dividend, omitting the place of units, and place the number (denoting how many) both in the quotient, and on the right of the divisor. VI. Multiply the divisor (thus augmented) by the figure last put in the quotient, and set the product under the dividend. VII. Subtract, and bring down the next period to the remainder for a dividend; and to the left of this bring down the last divisor with its right hand figure doubled, for a divisor. VIII. Find how often the divisor is contained in the dividend, omitting the units as before; put the number denoting how often in the quotient, and also on the right of the divisor. Multiply, subtract, bring down the next period, and also the divisor with its right hand figure doubled, &c. as before, and proceed in this manner till the work is finished. IX. If there is a remainder, periods of ciphers may be successively brought down, and the work continued as before, observing that the quotient figures which arise will be decimals; and if there be an odd decimal figure in the given number, a cipher must be subjoined, to make the right hand period complete. Also for every period of superfluous ciphers, either on the left or right of the given number, a cipher must be placed in the quotient. The operations may be proved by involving the root to the square, (Art. 265.) and adding in the remainder, if any.

ExAMPLEs. 1. Extract the square root of 54756.

- QPERATION. Erplanation. * . . . I first place a point over the units, then over \ 54756(234 = root the hundreds, then over the ten thousands; 5 4 being the first period, I find from the table the 43) 147 greatest square 4, contained in it; this 4 I place under the 5, and its root 2 in the quo129 tient, and having subtracted, I bring down to 464) 1856 the remainder 1 the next period 47, making * 1856 147 for the dividend; I double the quotient figure 2, and place the double, viz. 4, for a dia $ visor, to the left. Omitting the units 7, I ask * → Proof. how often 4 is contained in 14, and find it goes 3 times; this 3 I put both in the quotient and 234 = root divisor, making the latter 43; this I multiply by 234 = root the quotient figure 3, and subtract the product T936 129 from the dividend. To the remainder 18 I bring down the next period 56, making the new 702 dividend 1856; to the left of this I bring down 468 the divisor with its last figure 3 doubled, mak

R. Toro — ing 46: I then ask how often 46 goes in 185, 54756 = square. oiti. the 6,) it goes 4 times, I therefore put the 4 both in the quotient and on the right of the divisor, and multiply as before : there being neither a remainder, nor any more figures to bring down, the operation is finished,

2. Extract the square root of .000064807.

OPERATION".

~ 0000618076).008050279 root.

Erplanation.
Here, in order to complete the

| 64 right hand period, I subjoin a ci1605) 8070 pher; and there being 2 periods of 80.25 ciphers to the left, I prefix a cipher —- for each period to the root. In the 161002) 450000 second step, having brought down 322004 the 80, I find that 16 will not go in Toron 8; I therefore put a cipher both in 1610047) 14.99% the quotient i. . and then 11270329 bring down the next period 70; and

7100 the like in the next step. I bring

16100549) #. 1 down a period of ciphers both there

-- and in each following step. Remainder 8022 159

3. Extract the square root of 95.801234.
95.861°34(9.78781 root.
81
137) 1480
1309

1948) 17112
15584

19567) 152834
136969

195748) 1586500 1565984 1957561) 2051600 1957.561 Remainder 94039

4. Required the square root of 529. Root 23.
5. To find the square root of 104976. Root 324.
6. What is the square root of 298116 Root 546.

* These operations may be proved three ways. First, by involving the root to the square as in ex. 1. and adding the remainder to the square : the result, if the work be right, will equal the given number. Secondly, by casting out the nines: thus, cast out the nines from the root, and multiply the excess into itself; cast the nines out of the product, reserving the excess; cast the nines out of the remainder, subtract the excess from the dividend, and cast the nines out of what remains: if this excess equals the former, the work may (with the restriction mentioned in the note on Art. 41.) be presumed to be right. Thirdly, by addi. tion, similar to the proof of long division, Art. 41.

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7. Extract the square root of 974169. Root 987.

8. What is the square root of 10465292 Root 1023.

9. Extract the square root of 867.8916. Root 29.46. 10. Required the square root of 32.72869681. Root 5.7209. 11. Find the square root of 70. Root 8.3666, &c. 12. What is the square root of .000294? Root .0171464, &c. 13. What is the square root of 9892 Root 31.44837, &c. 14. Find the square root of 6.27. Root 2.50399.6805, &c. 15. Required y.00015241578750190521. Root .0123456789.

274. To extract the square root of a rulgar fraction, both terms of which are exact squares.

RULE. Extract the root of the numerator, and likewise of the denominator; these two roots will be respectively the terms of a new fraction, which will be the root required.

16. Extract the square root of To

Brplanation. Operation. Here the root of 4 is 2, and the root of

4 2 w/ 9 - #the Toot required. 9 is 3, therefore + will be the root of

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275. To extract the square root of a vulgar fraction, the terms of which are not both squares.

Rule. Reduce the given fraction to a decimal, (Art. 233.) and extract the root of this decimal for the answer.

1 18. What is the square root of +?

First # = .5 by Art. 233. Then v.5 = .70710678119, &c. (Art. 273.) the root required.

3 19. To find the square root of To Root .8660254, &c.

20. What is the square root of #? Root .2958.03989, &c.

276. To extract the square root of a mixed number.

Rule. Reduce the fraction to a decimal, (Art. 233.) to which prefix the whole number, extract the square root of the result by Art. 273. and it will be the root required.

21. To find the square root of 8 #. First (Art. 233) 8 # = 8.75.

Then (Art. 273.) v8.75 = 2.958.03989, the root required. l 2 22. What is the square root of i. ?

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Therefore v.344827586206, &c. = .5872202, &c. (Art. 273.) = the root required. 23. Required the square root of 14. Root 122474487, &c. 24. What is the square root of 7: 2 Root 2.792848, &c. - 24 25. What is the square root of 3i ? Root .8164965, &c. † 277. Sometimes it happens, that the given mixed mumber being reduced to its equivalent improper fraction, both the terms will be rational. In this case it will be best to extract the roots of the numerator and denominator separately, and they will form an improper fraction, which must be reduced to its proper terms, (Art. 173.)

26. Extract the square root of 2+. Thus (Art. 172.) 2, - +. Then (Art. 274) w/ = 14 (Art. 173) the root required.

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27. What is the square root of 2; ? Root 1;.
28. What is the square root of 30, 2 Root 54.
29. What is the square root of 11; ? Root 34.
30. Required the square root of 1564. Root 12}.

278. ProMiscuous Examples for PRActice.

1. The side of a square kitchen garden is 63 yards; how many. square yards does the garden contain Ans. 3969 square yards.

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