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The extraction of roots by feries is much facilitated by the binomial theorem (Chap. VI. Sect. 3.). By fimilar rules, founded on the fame principles, are the roots of numbers to be extracted.

III. Of Surds.

Def. Quantities with fractional exponents are called Surds, or Imperfect. Powers. Such quantities are alfo called irrational, in oppofition to others with integral exponents, which are called rational.

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

In the following operations, however, it is generally convenient to ufe the notation by the fractional exponents.



3 2

a3 = √ā. √4ab2=2ba3. “√a^b2=a*b3⁄4.


The operations concerning furds depend

on the followinging

principle. If the numera

tor and denominator of a fractional exponent

be both multiplied or both

d or both divided by the fame quantity, the value of the power is the





Thus, aan; for, let a=b;

then a"-b", and amc=bnc, and extracting

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

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

Rule. Reduce the fractional exponents to others of the fame value, and having the fame common denominator."


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a2=a2, and b3—b; therefore, and^

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Prob. II. To multiply and divide furds.

brs "d 1. When they are furds of the fame rational quantity, add and fubtract their exponents, {{

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2. If they are furds of different rational quantities, let them be brought to others of the fame denomination, if already they are not, by prob. 1. Then, by multiplying or dividing thefe rational quantities, their product or quotient may be fet under the common radical fign.



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Thus, √ax√ba"b" = √a^bm.

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If the furds have any rational coefficients, their product or quotient must be prefixed. Thus, a√n Xb√n=ab mn. It is often convenient, in the operations of this problem, not to bring the furds of fimple quantities to the fame denomination, but to exprets their product or quotient without the radical fign, in the fame manner as if they were rational quantities. Thus, the product in Ex. 1. may be amin, and the quotient in Ex. 3. at 1⁄2 bo.

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

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

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ayb=√a2b; 2×va =√8a.

Conv. ✔a2b3±abb; √4a*—8a3b=2a√1—26.

Even when the quantity under the radical fign is not divisible by a perfect power,


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

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Thus, yab-yaxNob, va b—bx = √ ba—b×× √a+x

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PROB. III. To involve or evolve furds.

This is performed by the fame 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 fame manner as to integers.


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