CASE VIII.

To inultiply fractional quantities together,

RULE.

Multiply the numerators together for a new numerator, and the denominators for a new denominator ;

and the former of these, being placed over the latter, will give the product of the fractions, as required (k).

EXAMPLES.

* X 2x

2x2

1018

2.C 1. It is required to find the product of and

6 9° Here

х2

the product required. 6 X9 54

27
%. It is required to find the continued product of

10.x
and
5 21

XX 4x X 10x . 40x3 4x 3
Here

the product.
2X5X21 210 21

ata 3. It is required to find the product of and

XX(a+x) 22 tax
Here

the product.
a xia-x) a?

)

a

a

(k) When the numerator of one of the fractions to be multiplied, and the denominator of the other, can be divided by some quantity which is common to each of them, the quotients may be used instead of the fractions themselves.

Also, when a fraction is to be multiplied by an integer, it is the same thing whether the numerator be multiplied by it, or the denominator divided by it. Or if an integer is to be multiplied by a fraction, or a fraction by an integer, the integer may be considered as having unity for its denominator, and the two be ther multiplied together as usual.

E

a

4. It is required to find the product of

3x 5х

and
2 36

2x 3x2
5. It is required to find the product of and

5 2a

2x 6. It is required to find the continued product of

3 4x2

and

atx 7. It is required to find the continued product of 2x 3ab

Бас
and

26
8. It is required to find the product of 2a+

bx

and 6 За

C

a

9. It is required to find the continued product Bag x+1

X-1 and 2a atb

10. It is required to find the continued product of a2 X2 al -62

and at a+b ax+x2'

ax

CASE IX,

To divide one fractional quantity by another.

RULE.

Multiply the denominator of the divisor by the numerator of the dividend, for the numerator ; and the nume. rator of the divicor by the denominator of the dividend, for the denominator. Or, which is more convenient in practice, multiply the dividend by the reciprocal of the divisor, and the product will be the quotient required. (1)

=1} Ans.

Here 2a

by

2x 1. It is required to divide. by

3 9 2x

9 9x 3 Here

X 3 9 3

22 6x

2

2a 40 2. It is required to divide by

b d d 2ad

ad X

Ans. b 40 4bc 2bc

x-+-6
3. It is required to divide eta

5x+a
sta 5ta 6x2 +6axta2
Here
X

Ans.
4
X-6 x+6 x2 - 62

2x2
4. It is required to divide by

a3 + x3 ta
2x2 ata_2x2 (x+a)

23 Here

X -a3 + x3 a(Q3 +23) xa--axta?

7x 3 5. It is required to divide

5

4.02 6. It is required to divide

7

by

by 5x

(1) When a fraction is to be divided by an integer, it is the same thing whether the numerator be divided by it, or the denominator multip'ied by it

Also, when the two numerators, or the two denominators, can be divided by some common quantity, that quantity may be thrown out of each, and the quotients used instead of the fractions first proposed.

2ax + x2 9. It is required to divide

by C3 - x3

at - 64 10. It is required to divide

332 – 2bx +62

INVOLUTION.

INVOLUTION is the raising of powers from any proposed root ; or the method of finding the square, cube, biquadrate, &c. of any given quantity.

RULE I.

> Multiply the index of the quantity by the index of the power to which it is to be raised, and the result will be the power required.

Or multiply the quantity into itself as many times less one as is denoted by the index of the power, and the last product will be the answer.

Note. When the sign of the root is t, all the powers of it will be t; and when the sign is ;

all the even pow. ers will be t, and the odd powers

-: as is evident from multiplication (m).

(m) Any power of the product of two or more quantities is equal to the same power of each of the factors multiplied together. And any power of a fraction is equal to the same power of the numerator divided by the like power of the denominator.

Also, am raised to the nth power is amn; end - a raised to the nath power is amn, according as n is an even or an odd number.

33-2axd tax

33 + 2ax? taas - ax+2ua2-a3

taxa + 2aoxtas X3–3ax: +3a-x-a3 cube. 1x3 +3ax: +3a2x ta3 cube.