9. Required the greatest common measure of the fraction x4 -- 3ax3 8a2x2 + 18a3x— 8aa

Ans. x2+2ax 2a2.

10. Required the greatest common measure of the fraction 5a5+10a2b+5a3b2

a3b+2a2b2 +2ab3 + ba°

Ans. a+b.

11. Required the greatest common measure of the fraction 6a5+15a1b-4a3c2. 10a2bc2

9a3b - 27a2bc

2

6abc2+18bc3

CASE II.

Ans. 3a2 2c2.

To reduce fractions to their lowest or most simple terms. RULE.-Divide the terms of the fraction by any number, or quantity, that will divide each of them without leaving a remainder; or find their greatest common measure, as in the last rule, by which divide both the numerator and denomina tor, and it will give the fraction required.

Whence c + x is the greatest common measure ;

and c + x

cx + x2

a2c + a2x

a2

to a single dimension only, I divide the same into the parts x2+2ax, and-bx-2ab; which, by inspection, appear to be equal to (x+2a)×x, and (x+2a) X-b. Therefore x+2a is a divisor to both the parts, and likewise to the whole, expressed by (x+2a) × (x —b); so that one of these two factors, if the fraction given can be reduced to lower terms, must also measure the numerator; but the former will be found to succeed, the quotient coming out x2 ax + bx — ab, exactly; whence the fraction itself is reduced to

22

— ax + bx— ab

-b X

which is not reducible farther by x-b, since the di

vision does not terminate without a remainder, as upon trial will be found. This rule is sometimes of great utility, because it spares great labour, and is very expeditious in reducing several fractions.-ED.

Whence a+bis the greatest common measure; and x+b)

b2x

x2+2bx+b2

bx

x2
x + b

the fraction required.

And the same answer would have been found, if x3 — b3x had been made the divisor instead of x2+2bx + b2.

To reduce a mixed quantity to an improper fraction. RULE.-Multiply the integral part by the denominator of the fraction, and to the product add the numerator, when it is

affirmative, or subtract it when negative; then the result, placed over the denominator, will give the improper fraction. required.

* xXx = x2. In adding the numerator a2-x2, the sign — affixed

denotes that the whole of that fraction is to be

subtracted, and consequently that the signs of each term of the numerator must be changed when it is combined with 22; hence the improper fraction is

x2-a2x2 2x2

a2

or

ર

ED.

To reduce an improper fraction to a whole or mixed quantity. RULE.-Divide the numerator by the denominator, for the integral part, and place the remainder, if any, over the denominator, for the fractional part; then the two, joined together, with the proper sign between them, will give the mixed quantity required.

ab - 2a2

to a mixed

ab

Ans. a+x+

3. It is required to reduce the fraction quantity.

4. It is required to reduce the fraction quantity.

5. It is required to reduce the fraction quantity.

6. It is required to reduce the fraction mixed quantity.

Ans. x2 + xy + y3

10x2

Ans. 2x1 +

5x*

CASE V.

To reduce fractions to other equivalent ones, that shall have a

common denominator.

RULE.-Multiply each of the numerators, separately, into

all the denominators, except its own, for the new numerators, and all the denominators together for a common denominator.*

EXAMPLES.

a

b

1. Reduce and to fractions that shall have a common

denominator.

b

bc the common denominator.

b

ас

с

2x

2. Reduce and to equivalent fractions having a com

α

*It may here be remarked, that if the numerator and denominator of a fraction be either both multiplied, or both divided, by the same number or quantity, its value will not be altered; thus

2 2 X 3 6

3

and 3X3 9

which method is often of great use in reducing fractions more readily to a common denominator.