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120. Multiplication of Quantities affected with Negative Expo
nents.—Suppose it is required to multiply a by
1 According to the preceding article, the result must be
1 But, according to Art. 76,
Hence we see that a-3 xa-2=a-6; that is, the rule of Art. 58 is general, and applies to negative as well as positive exponents.
Division of Fractions. 121. If the two fractions have the same denominator, then the quotient of the fractions will be the same as the quotient of their numerators. Thus it is plain that is contained in as often as 3 is contained in 9. If the two fractions have not the same denominator, we may perform the division after having first reduced them to a common denominator. Let it be required to divide by &
ad Reducing to a common denominator, we have to be di.
bd vided by
It is now plain that the quotient must be repre
bdo sented by the division of ad by be, which gives ad;
. a result which might have been obtained by inverting the terms of the divisor and multiplying by the resulting fraction; that is,
d ad b db bc
Hence we have the following
RULE. Invert the terms of the divisor, and multiply the dividend by the resulting fraction.
Entire and mixed quantities should first be reduced to fractional forms.
2c 1. Divide by
Ans. 13. 9
8. Divide 7a – 3x+- by 62.
21a-1-9nx +3m Ans.
10. Divide a?+ão
75cx- _ 3ab 9. Divide 15x2 - by
5cx-5a +56 23 ab
ax- _ boca + x3 by
2+y 12. Divide +by
+1 14. Divide x2 ++2 by æ+ão
a+b+c 15. Divide a’ – 22—4—2bc by
Ans. a -62 +c22ac.
122. Division of Quantities affected with Negative Exponents.
According to the preceding article, we have
1 a3 a3 1
1 CS a2 1
1 But, according to Art. 76, a5
a3 1 written a-3; and may
be written a-2, Hence we see that
a-5-4-3=a-2; that is, the rule of Art. 72 is general, and applies to negative as well as positive exponents.
Suppose it is required to divide
, may be written a-s; may be -9;
Ans. —a-3, or
EXAMPLES. 1. Divide a-5 by-a-? 2. Divide – a2 by a-'. 3. Divide 1 by a-4. 4. Divide 6an by – 2a-3,
5. Divide im- by bm.
123. The Reciprocal of a Fraction.—According to the defini. tion in Art. 34, the reciprocal of a quantity is the quotient arising from dividing a unit by that quantity. Hence the reciprocal of q is
7 = that is, the reciprocal of a fraction is the fraction inverted.
Thus the reciprocal of is 6#*; and the reciprocal of 1
is 6+c. btc
It is obvious that to divide by any quantity is the same as to multiply by its reciprocal, and to multiply by any quantity is the same as to divide by its reciprocal.
124. How to simplify Fractional Expressions.—The numerator or denominator of a fraction may be itself a fraction or a mixed
27 quantity, as In such cases we may regard the quantity
f above the line as a dividend, and the quantity below it as a divisor, and proceed according to Art. 121. Thus,
21- = x=10=31. The most complex fractions may be simplified by the application of similar principles.
bta a+6 This expression is equivalent to
bta or to
which is equal to me Ans. b