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.

3

Thus /3 or 3 denotes the square root of 3; 3 √7 or 74 denotes the cube root of 7; 5/21 or 21 denotes the fifth root of 21, &c.

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. . . . . . . 1. 4. 9. 16. 25. 36. 49. 64. 81. SQUARE ROOTS. 1. 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.

54756(234 root

4

43)147

129

464) 1856 し 1956

Proof.

Explanation.

I first place a point over the units, then over the hundreds, then over the ten thousands; being the first period, I find from the table the greatest square 4, contained in it; this 4 I place under the 5, and its root 2 in the quotient, and having subtracted, I bring down to the remainder 1 the next period 47, making 147 for the dividend; I double the quotient figure 2, and place the double, viz. 4, for a divisor, to the left. Omitting the units 7, I ask how often 4 is contained in 14, and find it goes 3 times; this 3 I put both in the quotient and divisor, making the latter 43; this I multiply by the quotient figure 3, and subtract the product 129 from the dividend. To the remainder 18 I bring down the next period 56, making the new dividend 1856; to the left of this I bring down the divisor with its last figure 3 doubled, making 46: I then ask how often 46 goes in 185, (omitting 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.

234 = root

234

root

936

702

468

·54756 square.

2. Extract the square root of .000064807.

OPERATION".

Explanation.

Here, in order to complete the right hand period, I subjoin a cipher; and there being 2 periods of ciphers to the left, I prefix a cipher for each period to the root. In the second step, having brought down the 80, I find that 16 will not go in 8; I therefore put a cipher both in the quotient and divisor, and then bring down the next period 70; and the like in the next step. I bring down a period of ciphers both there and in each following step.

3. Extract the square root of 95.801234.

95.801234(9.78781 root.

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.

:

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

8. What is the square root of 1046529 ? 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 989? Root 31.44837, &c. 14. Find the square root of 6.27. Root 2.503996805, &c. 15. Required .00015241578750190521. Root .0123456789.

274. To extract the square root of a vulgar 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.

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.

18. What is the square root of

1

1

?

First .5 by Art. 233. Then .5 .70710678119, &c.

2

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.

Then (Art. 273.) √/8.75 = 2.95803989, the root required.

22. What is the square root of

Thus (Art. 178.)

4층

ܕ܂

= .344827586206, &c.

Therefore 344827586206, &c. = .5872202, &c. (Art. 273.) the root required.

23. Required the square root of 14. 24. What is the square root of 73

?

21 25. What is the square root of ?

34

Root 1.22474487, &c.

Root 2.792848, &c.

Root .8164965, &c.

277. Sometimes it happens, that the given mixed number 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.)

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