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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 numbers to be extracted.

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III. Of Surds.

Def. Quantities with fractional exponents are called Surds, or Imperfeet. 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.

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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 fame quantity," the value of the power is the same. Thus, atranc; for, let añ=b then am=b", and ame=bac, and extracting

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

Thus, a=a= vai

a?b=a}b} =/ab.

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

Rule. Reduce the fractional exponents to

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


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Examples, was voor 97.0b; but ai ca, and bi=b*; therefore van and

od 77 are respectively equal to Vas and 54. angre od to silou gerti Prob. II. To multiply and divide surds,

noil }. When they are Jurds of the same rational

quantity, add and fubtract their exponents,

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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 set under the common radical sign.

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Thus, vāx vd=a"b" = Vabm.

b. & atb



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If the surds have any rational coefficients, their product or quotient must be prefixed. Thus, a 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 exprets 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 an in, and the quotient in Ex. 3. a

ats 65.

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


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



avbav.ab; 2x vā=va. Conv. Va?b3-abwb; v4a2—8a+b=2avi-26.

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

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

Thus, vabzvaxvby va bombx = vba-b** Vatim

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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 inte

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