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1 18. Multiply x2-x+1 by +
22 atba-6 462
a +6 19. Multiply
120. Multiplication of Quantities affected with Negative Expa
1 nents:—Suppose it is required to multiply as by
1 According to the preceding article, the result must be
1 But, according to Arl . 76, as may be written a-3
a-; written a-2; and of may be written as
Hence we see that a-3 xa-2=a-5; that is, the rule of Art. 58 is general, and applies to negative as well as positive exponents.
1 Ans. —-5, or
EXAMPLES. 1. Multiply — x-2 by x-3 2. Multiply a-2 by –a?. 3. Multiply a-3 by a3. 4. Multiply a-m by a". 5. Multiply a-m by a-n. 6. Multiply (a−b)" by (a–0)-3.
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 i by
ad Reducing to a common denominator, we have to be di.
. 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 d
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.
2x 1. Divide by
2a 2. Divide
75cx - 3ab 9. Divide 1522 - by x
Бcx — 5а+50" 2013 ab
axé _ boc2 + x3 10. Divide 2? + by a-b a-b
ab-am+bm 45(a−b) 27(a−b) 11. Divide
1 1 12. Divide
2+y + by
Ans. Unity. x+y y Y
x2 +1 14. Divide x2+z2+2 by ætä
a+b+c 15. Divide aș—62—4—2bc by
Ans. al -52 +02-2ac.
y2 y У
122. Division of Quantities affected with Negative Exponents.
Suppose it is required to divide a by According to the
preceding article, we have
1 a3 a3 1
1 written a-3; and be written a-2. Hence we see that
a-5-a-3=a-2; that is, the rule of Art. 72 is general, and applies to negative as well as positive exponents.
But, according to Art
. 76, may be written a–; 1
1 Ans. -a-3, or
EXAMPLES. 1. Divide a-s by-a-? 2. Divide -a? by a-. 3. Divide 1 by a-4 4. Divide 6an by – 2a-3.
5. Divide bm- by bm.
123. The Reciprocal of a Fraction.-According to the definition in Art. 34, the reciprocal of a quantity is the quotient arising from dividing a unit by that quantity. Hence the reciprocal of s is
1-6=1x that is, the reciprocal of a fraction is the fraction inverted.
Thus the reciprocal of 64.2 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 above the line as a dividend, and the quantity below it as a divisor, and proceed according to Art. 121. Thus,
21- =*==3}. The most complex fractions may be simplified by the application of similar principles.
bta atb. This expression is equivalent to bha
b+a or to
which is equal to ģ, Ans.