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the root of the coefficient may be extract

І І І я ed.

U a Thus, the square root of a*=a*=#a?,

?? myo *s 25 270o=393=za 3

Va4612 =a* 6 #abs. 2. The exponents of the letters may not be multiples of the index of the root, and then they become fractions; and when the root of the coefficient cannot be extracted, it may also be expressed by a fractional exponent, its original index being understood

to be I.

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Thus, V16ab3=4a3b

* 17ax?=73a-x='Vāxa}x.

= As evolution is the reverse of involution, the reason of the rule is evident.

The root of any fraction is found by extracting that root out of both numerator and denominator.

Case II. When the quantity is compound. 1. To extract the square root.ca

RULE

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1. The given quantity is to.

tity is to be ranged

ranged according to the powers of the letters, as in division,

Thus, in the example a2 + 2ab+b?, the quantities are ranged in this manner. So

Vita 2. The Square root is to be extracted out of

the first term (by preceding rules), which gives the first part of the root fought. Subtract its square from the given quan, tity, and divide the first term of the remainder by double the part already found, and the quotient is the second term of the root.

Thus, in this example, the remainder is 2ab +62; and 2ab being divided by za, the double of the part found, gives + b for the second part of the root.

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3. Add this second part to double of the first, and multiply their sum by-the fecond part:

Subtract

Subtract the product from the last remainder, and if nothing remain, the square root is obtained. But, if there is a remainder, it muft be divided by the double of the

parts already found, and the quotient will 1 give the third part of the root; and foon.

In the last example, it is obvious that 6+6 is the square root fought.:

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The reason of this rule appears from the composition of a square idtun ti bore ish

sisisi zi studi u tu boiuidi zs
su u 2. To extract tanývother roote iz
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scht heo has coauthor and Rule. Range the quantity according to the

dimensions of its letters, and extract the faid root out of the firf term, and that Shall be the firft member of the root required. Then raise this root to a dimenson lower by anit, than the number that denominates the root required, and multiply the power that arise's, by that number itself : divide the second term of the given quantity by the product, and the quotient. Shall give the second member of the root required. In-like-manner are the other parts to be found, by considering those already got as making one term.

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Thus, the fifth root of

as+5a4b+10a%b2 +10ab? +5ab4 +-b}(a+b

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And a+raised to the 5th power is the given quantity, and therefore it is the root sought.

In evolution 'it will often happen, that the operation will not terminate, and the root will be expressed by a series.

Thus, the square root of a? +*2 becomes a series.

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