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2a 1. Required to find the product of -and

8 5 a x2а 2a2

a 2 Here

the product required. 8 X 5 40

3a 2. Required the product of and


7 axЗа хба 1823

the product required. 3 X 4 X 7


a + b 3. Required the product of - and

6 2a + c 20 x (a+b) 2aa + 216 Here

the product required. O X (20+0) 2ab + bc

4a ба 4. Required the product of - and

3 50

3a 452 5. Required the product of -and

За 3a 8ас 4ab 6. To multiply and and together. b 6 30

ab 3a 7. Required the product of 2a + -- and

20 6

2a? - 262 422 + 262 8. Required the product of


a +6

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

2a + 6 x2 10. Multiply at by x +



CASE IX. To Divide one Fractional Quantity by another. DIVIDE the numerators by each other, and the denomina. tors by each other, if they will exactly divide. But, if not, then invert the terins of the divisor, and multiply by it exactly as in multiplication,

EXAMPLES. 1 If the fractions to be divided have a c.mmon denominator, take the nurierator of the divider.d for a n w numerator, and the numerator or the divisor:or the new denominator.

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

8. When



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a + b


a 3a
1. Required to divide-by-.


2 Here


- the quotient.
4 8



За 50 2. Required to divide


26 4d 3a 50 За 40


bad Here


--the quotient. 26 4d 26 50 10bc 5bc

2a + 6 3a + 26 3. To divide by

Here, 32-26 4a + b 2a + b 40 + 6

8a2 + 6ab + 6% Х

the quotient required. 39-26 3a +26


462 3a2

2a 4. To divide

al + 62

2a + 26
3a? x (a + b)

3a Here,

Х al + 63 @ (a3 + 63) xa a> - ab + be is the quotient required

3.2 1 5. To divide - by


6. To divide by 3.3.


36 +1 41 7. To divide by



42 8. To divide

2x 1 3

3a 9. To divide - by

5 56 2a-6

5ac 10. To divide by

4cd 6d

524-504 692 + 5ab 11. Divide

2a2 4a6 + 262


INVOLU3. 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 proposed.

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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 t, 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. a, the root

a', the root = square.

square = cube


cube 4th power

4th power 25 = 5th power


= 5th power &c.


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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 ex. ponents.

a3 Thus, as X = R3+= 45. And ad- a2 or-sa

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

EXAMPLES FOR PRACTICE. 1. Required the cube or 3d power of 3a2, 2. Required the 4th power of 2a2b. 3. Required the 3d power of — fa?b3. 4. To find the biquadrate of

262 5. Required the 5th power of a -2x. 6. To find the 6th power of 22.


SIR ISAAC NEWTON'S RULE for raising a Binomial to any.

Power whatever*.

1. To find the terms without the Co-eficients. 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 ad or following quantity, the indices of the terms are 0, 1, 2, 3, 4, &c increasing always by i. That is, the first term will contain only the 1st part of the root with the same index, or of

1 - 1



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

7-11-2 (a + x)n = + n.2h-2xtn.

22-232 + n.

-an-3*? &c. 2

2 3

-2 (a-)n = ab – n.06-1x tn. 20-%

-n-3x3&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 + power : 1+ 2+1-4 = 22 in the square ; 1 + 3 + 3 + 1 = 80 23 in the cube, or third power; and so on,


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2 in the first

the same height as the intended power : and the last term of the series will contain only the 2d part of the given root, when raised also to the same height of the intended power: but all the other or intermediate terms will contain the products of some powers of both the members of the root, in such sort, that the powers or indices of the 1st or leading member will always decrease by 1, while those of the 2d member always increase hy l.

2. To find ihe Co-efficients. The first co-efficient is always 1, and the second is the same as the index of the intended power; to find the 3d co-efficient multiply that of the 2d term by the index of the leading letter in the same term, and divide the product by 2 ; and so on, that is, multiply the coefficient of the term last found by the index of the leading quantity in that term, and divide the product by the number of terms to that place, and it will give the co-efficient of the term next following ; which rule will find all the co-efficients, one after another.

Note. The whole number of terms will be 1 more than the index of the given power: and when both terms of the root are +, all the terms of the power will be +'; but if the se. cond teren be -, all the odd terms will be +, and all the even terms which causes the terms to be + and alternaiely. Also the sum of the two indices, in each term, is always the same number, viz. the index of the required power : and counting from the middle of the series, both ways, or towards the right and left, the indices of the two terms are the same figures at equal distances, but mutually changed places. Moreover, the co-efficients are the same numbers at equal distances from the middle of the series, towards the right and left; so by whatever numbers they increase to the middle, by the same in the reverse order they decrease to the end.


1. Leta to be involved to the 5th power.

The terms without the co-efficients, by the 1st rule, will be

as, aix, ar?, ar3, ar*, rs,
and the co-efficients, by the 2d rule, will be

5 X 4 10 X 3.10 X 25 X 1
1, 5,

i 2 3

or 1, 5, 10, 10, 5, 1;

Therefore the 5th power altogether is
as + 5a*r + 10a3.r? + 10a233 + 5ax* + x6.
Vou. I.


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