and these latter accord with the rules for positive exponents. In like manner, with positive exponents, we have

and with negative exponents for the reciprocals, we have

xTM×x ̄ˆ=xm—”, x”÷x ̄”—x2+”,and x¬”÷÷x”—x¬m—n ; which latter equations are those which arise from applying the rules for multiplication and division, without any restriction as to the signs of the exponents.

It thus appears that provided * always stand for the

rules for multiplication and division apply generally whether the exponents be whole or fractional, positive or negative.

It follows from what has now been shown that

xm

хт

xm

but the first member is 1; therefore, x=1, whatever be x; that is, in the notation for exponents, the symbol x is the representative of unity.

One more particular remains to be noticed. It has already

m

been explained that a means the nth root of the mth power of a; it equally means the mth power of the nth root of a; for the two

results are identical; that is to say, (a)"is the same as

For the former is a" an an an to m factors;

(am).

and the latter is the nth root of aa aa...... to m factors;

1 1 1

which is an an añ an ...... to m factors, as before.

NOTE. That a fractional exponent may be reduced to its lowest terms without

root of the r" power :-two neutralising operations, leaving the thing operated upon

untouched: hence, anr is the same as a ".

64. Having thus established the universality of the rules for multiplication and division, we shall proceed to apply them to a

few examples in which the restriction hitherto imposed, as to the character of the exponents, is removed.

(1) Multiply by x-y.

3+x23⁄4/y+√x33⁄4/y2+xy+√/x.3⁄4/y*+Vy5

Changing the notation, using exponents throughout instead of radical signs, we have

therefore the quotient, using only positive exponents, is

(3) Divide 3x2-26xy+35y2 by x*-7x-3y

EXAMPLES FOR EXERCISE.

(1) Multiply x2+x2+1 by x3—1.

(2) Multiply a+a*b*+a*b*+b* by a1_b1.

(3) Multiply a—xy+x*y*—y2 by x2+y2.

(4) Multiply 15-6-7x-5+6x by 82-2-3x-1. (5) Multiply a2-b by a -b

(8) Divide x3+2xy + 2x3y+y* by x*+y*.

(9) Divide x1-y-1 by x -y

(10) Divide x *_* by x }_—y3.

(11) Divide xy-1+2x13 y ̄ ̄1+ 3 + 2 x ̄ ̄ ̄1 y3 + x1y by

(12) Divide x

-3x3y3+5y—3x ̄13y3+x-'' by the square

REDUCTION OF SURDS.

65. The examples in the preceding article have been given here for the purpose of completing algebraical multiplication and division, and of showing that the rules are the same under every supposition as to the character of the exponents connected with the quantities concerned; but surd or irrational expressions are sometimes presented in a form which admits of simplifications; and the reduction, necessary for this purpose, as well as some other particulars belonging to the management of surds, are now to be considered.

66. To reduce a rational quantity to the form of a surd:RULE.-Raise the rational quantity to the power denoted by the index of the proposed surd; then the corresponding root of this power, expressed by means of either the radical sign or a fractional exponent, will be the given quantity under the proposed form.

Thus 2, reduced to the form of the square root, is √4; a2 reduced to the form of the cube root, is ao, or a3, 3x reduced

to the form of the fourth root is 81a, or (81x1)*, or (3*x1)*;

and so on.

This reduction of a rational quantity to the form of a surd may seem to the learner to be nothing more than the conversion of a simple into a complicated form, and to be little likely to facilitate algebraic operations; but this would be an erroneous idea. Suppose, for example, we had to multiply

Xa x+a by

x+a X-a

If the second factor is to preserve its rational form, the expres

the rational quantity to the form of the square root, it becomes

changed into

(x+a)
(x-a)'

=

which is the simpler form. Other instances

of such simplifications will have been met with in the solutions to examples already given; for the reduction here spoken of is so obviously allowable, that we have not scrupled to apply it occasionally in the preceding parts of the work: it is formally introduced here only to give the greater completeness to the subject of surds.

67. To reduce surds to their most simple forms.

In order that a surd may admit of simplification, it must be decomposable into two factors, one of which will be a perfect power corresponding to the surd root; hence, to simplify such surds, the following is the rule, namely:

RULE.-Multiply the root of that factor which is the complete power, by the other factor with the proper radical sign or index connected with it.

If the surd be in the form of a fraction, the simplification may always be effected by first multiplying the terms of the fraction by such a quantity as will convert the denominator into a power corresponding to the surd root; the factors of the quantity under the surd index will then be the numerator of the changed fraction, and 1 divided by the denominator: the latter being the perfect power. The following examples will make what is here said sufficiently plain.

(1) Reduce 320 to its most simple form.

Here the highest factor of 320, which is a square, is 64, the other factor being 5; hence √320=√64 × √5=8√5.

(2) Reduce

to its most simple form.

Here the denominator 16 is 42; hence, multiplying the terms

of the fraction by 4, the expression becomes

28

=28.

43

NOTE.-It is proper to remark here that we consider a surd to be reduced to its most simple form when the quantity left under the radical is the smallest integer possible; the superior simplicity, however, except in the case of a fractional surd, and sometimes even then, is often more in appearance merely than in reality. In the first of these examples, for instance, 8/5 is not in strictness a more simple expression than 320; it is in fact rather less so; for there is as much trouble incurred in extracting the square root of 5 to a given extent of figures, as in extracting the square root of 320; and after the cxtraction of the root of the former number a multiplication of that root by 8 still remains to be performed. In the second example, however, which is a case of a fractional surd, the change effected contributes to arithmetical facility; for in the form as originally proposed, the calculation of the surd would require that we divide 7 by 16, take the cube root of the quotient, and then divide that root by 2; in the changed form, no division, previously to extraction, is required; we merely have to take the cube root of 28, and to divide that root by 8.

Tables of square and cube roots are of great service in such operations as those just noticed; but the remarks here made, in reference to the comparative simplicity of the original and the so-called simplified forms of the surd, equally apply whether the roots are actually extracted, by the usual arithmetical processes, or are at once taken out of a table.*

(3) Reduce

a+b

to its most simple form. a-b

Multiplying numerator and denominator by a-b, the surd

*The most useful set of tables, for such purposes as those noticed in the text are "Barlow's Tables of Squares, Cubes, Square-roots, Cube-roots, and Reciprocals of all integer numbers up to 10,000." These tables, edited and corrected by Professor De Morgan, have been published in a very convenient form by Taylor and Walton: London, 1840.