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Ex. 5. Find the square root of 214369.

Ans. 463.

Ex. 6. Find the square root of 393129.

Ans. 627.

Ex. 7. Find the square root of 758641.

Ans. 871.

(140.) If, after all the periods have been brought down, there is no remainder, the proposed number is a perfect square. But if there is a remainder, we have

only found the entire part of the root. In this case ciphers may be annexed forming new periods, each of which will give one decimal place in the root.

Ex. 8. Find the square root of 2972.

The operation is as follows:

29.72 54.516+

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Consequently, the square root of 2972 is 54.516, with a remainder of 5744. By annexing a greater number of ciphers, the root may be obtained to a greater number of decimal places; but, however far the operation may be carried, we shall always find a remainder.

QUEST. What must be done when there is a remainder after all the periods have been brought down?

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Ex. 9. Find the square root of 929296.

Ans. 964.

Ex. 10. Find the square root of 18957316.

Ans. 4354.

Ex. 11. Find the square root of 47348161.

Ans. 6881.

Ex. 12. Find the square root of 77158656.

Ans. 8784.

Ex. 13. Find the square root of 88078225.

Ex. 14. Find the square root of 87.

Ans. 9385.

Ans. 9.3273+.

Ex. 15. Find the square root of 158.

Ans. 12.5698+.

Ex. 16. Find the square root of 523.

Ans. 22.8691+.

Ex. 17. Find the square root of 654.

Ans. 25.5734+.

Ex. 18. Find the square root of 763.

Ans. 27.6224+.

Ex. 19. Find the square root of 2.

Ans. 1.4142+.

Ex. 20. Find the square root of 3.

Ans. 1.7320+.

PROBLEM II.

To Extract the Square Root of Fractions. (141.) The second power of a fraction is obtained by multiplying the numerator into itself, and the de

nominator into itself. Thus the second power of

QUEST.-How do we obtain the square root of a fraction?

a

is b

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Hence the square root of a fraction is equal to the square root of the numerator divided by the square root of the denominator.

4

Ex. 1. What is the square root of?

Ans.

9

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4,

(142.) If either the numerator or denominator is not a perfect square, we may change the vulgar frac tion into a decimal, continuing the division until the QUEST.-When the terms of the fraction are not perfect squares, how must we proceed?

double the number of Then extract the root

number of decimal places is
places required in the root.
of the decimal fraction by Art. 140.

6

Ex. 7. What is the square root of ?

6

17

The fraction reduced to a decimal fraction, is

17'

35294117; the square root of which is .59408 Ans.

Ex. 8. What is the square root of?

3

7

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(143.) The square root of a mixed quantity may be found in the usual way, if we reduce the vulgar fraction to its equivalent decimal, and divide the number into periods commencing with the decimal point. Ex. 10. What is the square root of 3?

Ans. 1.8257.

Ex. 11. What is the square root of 101?

Ans. 3.2015.

Ans. 2.6832.

Ex. 12. What is the square root of 7?

Ex. 13. What is the square root of 294?

Ans. 5.3984.

Ex. 14. What is the square root of 32.462.

Ans. 5.6975.

Ex. 15. What is the square root of 75% ?

QUEST.-HOW may we find the square root of a mixed quantity?

Ans. 8.6849.

Ex. 16. What is the square root of 57?

Ans. 2.3629.

3

Ex. 17. What is the square root of?

Ans. .77459.

2

Ex. 18. What is the square root of?

3

Ans. .81649.

Ans. 7.656.

Ex. 19. What is the square root of 58.614336?

Ex. 20. What is the square root of 9.878449 ? Ans. 3.143.

PROBLEM III.

To Extract the Square Root of Monomials. (144.) According to Art. 120, in order to square a monomial, we must square its coefficient and multiply the exponent of each of its letters by 2. Hence, in order to derive the square root of a monomial from its square, we have the following

RULE.

1. Extract the square root of its coefficient. 2. Divide the exponent of each letter by 2. Ex. 1. Thus √64a*b*=8a'b.

This is evidently the true result for

(8a2b)'=8a'bx8a'b=64a*b*.

Ex. 2. Find the square root of 81a*b*.

Ans. 9a b3.

Ans. 15a'b'c.

Ex, 3. Find the square root of 225a*b*c2.

QUEST.-How do we extract the square root of a monomial

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