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The extraction of roots by series is much
facilitated by the binomial theorem (Chap:
VI. Sect. 3.). By similar rules, founded on

. )
the same principles, are the roots of num-
bers to be extracted.

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Def. Quantities with fractional exponents are called Surds, or Imperfect. Powers.

Such quantities are also called irrational, in opposition to others with integral exponents, which are called rational.

Surds may be expressed either by the fractional exponents, or by the radical fign, the denominator of the fraction being its index; and hence the orders of surds are denominated from this index.

In the following operations, however, it
is generally convenient to use the notation,
by the fractional exponents.
at='va. 1720P=2ba? Vab=a*b*.
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The operations concerning surds depend on the following principle. If the numerator and denominator of a fractional exponent be both multiplied or both divided by the same quantity, the value of the power is the same. Thus, ama

, an zane

an=anc; for, 'let =b; then a"=b", and ame=b",

=bnc, and extracting the root nc, auc=bc=b=an.

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Lem. A rational quantity may be put into the form of a surd, by reducing its index to the form of a fraction of the fame value.

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Prob. I. To reduce surds of different denominators to others of the same value, and of the fame denomination,

Rule. Reduce the fractional exponents to

others of the same value, and having the Jame common denominator.

Example.

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sprit sinnsy102 Example., Vaivora, but

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C13 aizal, and bi=65; therefore van and F are respectively equal to Vand Vo

54 ads to sino ads pisaus To multiply and divide surds,

fpj Į. When they are surds of the same rational

quantity, add and subtract their exponents, its

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Prob. II.

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2. If they are surds of different rational

quantities, let them be brought to others of the same denomination, if already they are not, by prob. 1. Then, by multiplying or dividing these rational quantities, their product or quotient may be fet under the common radical sign.

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1

mn

Thus, Vaxh = ambn = Vabm.
Х

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V

= Vaab. V atb

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atb

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If the surds have any rational coefficients, their product or quotient must be prefixed. Thus, a v ñ xbvin=ab v mn. It is often convenient, in the operations of this problem, not to bring the surds of simple quantities to the same denomination, but to express their product or quotient without the radical sign, in the same manner as if they were rational quantities. Thus, the product in Ex. 1. may be a min, and the quotient in Ex. 3. a7365.

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Cor. If a rational coefficient be prefixed to a radical sign, it may be reduced to the form of a surd by the lemma, and multiplied by this problem ; and conversely, if the quantity under the radical sign be divisible by a perfect power of the same de

nomination,

1

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nomination, it may be taken out, and its root prefixed as a coefficient.

3

3

avb=V.ab; 2x va=v8a.
Conv. Vab3 aby b; v4a2-8a+b=2avi-26.

Even when the quantity under the radi-
cal sign is not divisible by a perfect power,
it
may

be useful fometimes to divide furds into their component factors, by reversing the operation of this problem.

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3

3

Thus, vabavax või va b-b* = vba-b** vat*
=xx
67 x vax* va+x.

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PROB. III. • To involve or evolve surds.

This is performed by the same rules as in other quantities, by multiplying or dividing their exponents by the index of the power or root required.

The notation by negative exponents mentioned in the lemma at the beginning of this chapter, is applicable to fractional exponents, in the same manner as to integers.

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