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merator of the divisor by the denominator of the dividend for a new denominator.

Or, invert the terms of the divisor, and then multiply by it, exactly as in multiplication:

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Here *2 {=}} is the quotient required, 2. Divide za by Cure

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7. Divide

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ty

INVOLUTION is the continual multiplication of a quäntie

into itself, and the products thence arising are called the powers of that quantity, and the quantity itself is called the root. Or it is the method of finding the square, cube, biquadrate, &c. of any given quantity,

RULE.*

Multiply the quantity into itself, till the quantity be taken for a factor as many times as there are units in the index, and the last product will be the power required.

Or, Multiply the index of the quantity by the index of the power, and the result will be the power required.

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Any power of the product of two or more quantities is equal to the same powers of the factors, multiplied together.

And any power of a fraction is equal to the same power of the numerator, divided by the same power of the denominator,

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*4+39x3 +3a%x* tax
+ ax +39***

t3a3x tă
*4+42x3+62*x* +4a3x +a*

= 4th power,

The third power of x* is ***3, or **.
The fourth power of 2a3b* is 24 Xü'*b8, or 16a" * }*.
The mth power of q") is apg",
The second power of extis.cx?**?; or axt, that is,

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is a: +xa1

3 m

and the mth power of a' txal

*** txi, namely, the nth power of the cube root of ****:

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Note. All the odd powers raised from a negative root are negative, and all the even powers are positive.

Thus, the second power of a is - -axmasta? by the rule for the signs in multiplication.

The third power of --a is ta' x-s.
The fourth power is -a: Xa=tat.
The fifth power of

ma is t-a4 x=5, &c.

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EXAMPLES FOR PRACTICE, 1. Required the cube of ---8x*y, Ans. -- 512x*y?

2a x 2. Required the biquadrate of

361

1698x4 Ans.

8168 3. Required the 5th power of amenitie Ans. aS

--p*x+104*** 109*** +59x*_**

SIR ISAAC NEWTON'S RULE For raising a binomial or residual quantity to any power

whatever, *

1. To find the terms without the coefficients. The index of the first, or leading quantity, begins with that of the

given

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1-I

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2

* This role, expressed in general terms, is as follows : a+b =a" tn. ath? 6+

262 tn

3 ??, &c.

2

gery'

es

given power, ad decreases continually by i, in
term to the last"; and in the following quantity the inu
of the terms are 0, 1, 2, 3, 4, &c.

2. To find the unicè òr coefficients. The first is always i, and the second is the index of the power; and in general, if the coefficient of any term be multiplied by the index of the leading quantity, and the product be divided by the number of terms to that place, it will give the coefficient of the term next following:

Note. The whole number of terms will be one more than the index of the given power; and, when both terms of the root are t, all the terms of the power will be t ; but if the second term be then all the odd terms will be +, and the even terms

EXAMPLES

1. Let a-t* be involved to the fifth power.

The terms without the coefficients will be
as, a*x, aix', 'a'xs, ax", %5;

and the coefficients will be

IOX2 5

5X1 1, 5

SX
3

4 5
or 1, 5, 10, 10, 5, 1;
And therefore the 5th power is
S+50*x+-10aox' +100*43+5ax*-+*s.

2. Let

en

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2

2

nii
amb 34"-1. an-b71. 4926

3
1}, &c.

Note. The sum of the coefficients, in every power, is equal to the number 2, raised to that power. Thus, 1+1=2, for the first power ; 1+2+1=4=2', for the square ; 1+3+3+1= 832', for the cube, or third power; and so on.

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