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30 4a

58 Here 219 16a

is the difference required. 4 28 28 28

2a 6 30 - 46 2. To find the difference of

and 4c

3b

b

2a Here

46

3a - 46 6ab- 3bb

3b 12bc

12ac 16bc

12bc

6ab- 366 – 12ac + 16bc

is the difference required. 12bc

10a

4a 3. Required the difference of and

3a 4. Required the difference of 6e and

4

5a 20 5. Required the difference of and

4

3 28

3a + c 6. Subtract from

b 2a +6 4a + 8 7. Take from 9

5 36

2a 8. Take 2a

from 4a +

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20

с

CASE VIII.

To Multiply Fractional Quantities together. MULTIPLY the numerators together for a new numerator, and the denominators for a new denominator *.

*

1. When the numerator of one fraction, and the denominator of the other, can be divided by some quantity, which is common to both, the quotients may be used instead of them.

2. When a fraction is to be multiplied by an integer, the product is found either by multiplying the numerator, or dividing the denominator by it; and if the integer be the same with the denominator, the numerator may be taken for the product.

EXAMPLES.

EXAMPLES.

5

a

ва

and

a + b

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20 1. Required to find the product of g and a X 2a

2a? a? Here

the product required. 8 x 5 40 20

3а 2. Required the product of and

34 7 @ X 3a x 60 18a3

303

the product required. 3 X 4 X 7 84 14

2a 3. Required the product of

6

2a +6
2a x (a+b)
Here

2aa + 2ab

the product required. bx (2a + c) 2ab + bc

4a 4. Required the product of and

5c 3a

462 5. Required the product of and

4 3a 3a

Sac 4ab 6. To multiply

and and together. 62 b

30

ab 7. Required the product of 2a + and

2c

b

2a? 262 4a+ 232 8. Required the product of

and Sbc

a + 6

2a 1 9. Required the product of 3a, and 2a + 1

and

2a +3 22

a? 10. Multiply at

by 2 2a 4a?

3a?

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+

2.2

4.22

CASE IX.

Ta Divide one Fractional Quantity by another. Divide the numerátors by each other, and the denominators by each other, if they will exactly divide. But, if not, then invert the terms of the divisor, and multiply by it exactly as in multiplication*.

EXAMPLES.

* 1. If the fractions to be divided bave a common denominator, take the numerator of the dividend for a new numerator, an the numerator of the divisor for the new denominator.

2. When

EXAMPLES.

1. Required to divide by

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az + ze by

a+b

8 Sa

8 8a 2 Here

Х

the quotient.
8
За 12a

3

3a 50 2. Required to divide

2b
by

4d
32 50 3a 4d 12ad 6ad
Here

the quotient.
26
4d

Х
26 50 .10bc 5bc

2a + b 3a + 2b 3. To divide by

stere, 30-26 4a + 26 2a + b 4a + b 8a + ab + 32 Х

the quotient required. 3a - 26 3a+2b 9a? – 4.62

3a?

2a 4. To divide

2a +26 3a?

3ax (a + b) За Here,

Х a + 63

(a3 + b3) xa a2 - abt to is the quotient required.

3.7 11 5. To divide

by
4 12

6x2 6. To divide

5

3x + 1 4.7 7. To divide by

9

3

4.10 8. To divide

2.0 1

4x За 9. To divide

by 5

56° 2a-6

5ac 10. To divide

4cd

5a4-564 6a2 + 5ab 11. Divide

by 2a?

4ab + 262 4a 46

a

by 3x.

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by

by 6d

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2. When a fraction is to be divided by any quantity, it is the same thing whether the numerator be divided by it, or the denominator multiplied by it.

3. When the two numerators, or the two denominators, can be divided by some common quantity, let that be done, and the quotients used instead of the fractions first

INVOLUTION.

proposed.

INVOLUTION. INVOLUTION is the raising of powers from any proposed root; such as finding the square, cube, biquadrate, &c, of any given quantity. The method is as follows:

MULTIPLY the root or given quantity by itself, as many times as there are units in the index less one, and the last

product will be the power required.-Or, in literals, multiply the index of the root by the index of the power, and the result will be the power, the same as before.

Note. When the sign of the root is +, all the powers of it will be + ; but when the sign is –,

all the even powers will be t, and all the odd powers

- ; as is evident from multiplication.

EXAMPLES.

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* 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, powers or roots of the same quantity, are multiplied by one another, by adding their exponents; or divided, by subtracting their exponents.

Thus, a' x a' @+a'. And a = or

3-2

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a.

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the cubes, or third powers, of x -a and x ta.

EXAMPLES FOR PRACTICE. 1. Required the cube or 3d power of 3a*. 2. Required the 4th power of 2a*b. 3. Required the 3d power of -4a2b3.

a x 4. To find the biquadrate of

262 5. Required the 5th power of a - 28.

6. To find the 6th power of 2a*. Sir Isaac Newton's RULE for raising a Binomial to any

Power whatever * 1. To find the Terms without the Co-efficients. The index of the first, or leading quantity, begins

with the index of the given power, and in the succeeding terms decreases continually by 1, in every term to the last; and in the 2d or following quantity, the indices of the terms are 0, 1, 2, 3, 4, &c, increasing always by 1. That is, the first term will contain only the 1st part of the root with the same index, or of

* This rule, expressed in general terms, is as follows :

n-1

In -2 a-x= -12 + n

an222 + n.

-an-323&c. 2

3 n-1

n-ln-2 (a-x)"a"-na-), + n. Q9-%%.

333 &c. 2

2 3 Note. The sum of the co-efficients, in every power, is equal to the number 2, when raised to that power. Thus 1+1 = 2 in the first power; 1+2+1=4 = 22 in the square; 1 +3+3 .+1=8 = 23 in the cube, or third power ; and so on.

the

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