Imágenes de páginas
PDF
EPUB
[merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][ocr errors][merged small]

120. Multiplication of Quantities affected with Negative Expo

1

nents.—Suppose it is required to multiply a by

[ocr errors]
[ocr errors]

,

[ocr errors]

may be

1 According to the preceding article, the result must be

a5

1 But, according to Art. 76,

may
be written a-3;

a2
1
written a-2; and e may be written a-5.

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.

= a

[merged small][ocr errors][merged small][ocr errors][merged small]

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 &

bc

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

a

с

a
Х

[ocr errors]

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.

EXAMPLES.

2c 1. Divide by

Ans. 13. 9

[ocr errors]
[ocr errors]
[merged small][merged small][merged small][ocr errors][ocr errors][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small]

8. Divide 7a – 3x+- by 62.

[ocr errors]

n

21a-1-9nx +3m Ans.

36212

— аг

[ocr errors]
[ocr errors]
[ocr errors]

с

10. Divide a?+ão

m.

a-b

+

3ab
a-6

75cx- _ 3ab 9. Divide 15x2 - by

Ans. 5c

5cx-5a +56 23 ab

ax- _ boca + x3 by

Ans. a

ab-am+bm
45(ab) 27(a-6)
11. Divide

by
32(m+b) 128n(m+b)
22
1
1 1

2+y 12. Divide +by

Ans.
yo
y2 y

y
30+2y
13. Divide

Ans. Unity
x+y y Y a+y
1
1

+1 14. Divide x2 ++2 by æ+ão

5

a+b+c 15. Divide a’ – 22—4—2bc by

a+b-c

Ans. a -62 +c22ac.

[ocr errors]

x+2y +

[ocr errors]
[ocr errors]

by

Ans. 22+1

[ocr errors]
[ocr errors]
[ocr errors]

Х

122. Division of Quantities affected with Negative Exponents.

1 1
by
ago

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

2x

, may be written a-s; may be -9;

.

Ans. —a-3, or

1
as

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.
6. Divide 12x-2y4 by - 4xy?.
7. Divide (-y)-4 by (oc-y)-6.

a

[ocr errors]

a

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.

a

[ocr errors]

a

a

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.

[merged small][merged small][merged small][ocr errors][merged small]

bta a+6 This expression is equivalent to

bta or to

Х

which is equal to me Ans. b

[ocr errors]

a

a

a+6

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small]
[ocr errors]

35

Ans. 4.

3. Find the value of the fraction

23

22 abc

39mna 4. Find the value of the fraction

11ab

3mx a+ba-6

it c+d

d 5. Simplify

atb, a-6

it
C-d'ctd

[ocr errors]

ac-bd Ans.

ac+bd

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]
« AnteriorContinuar »