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Then
First Sup.

Second Sup.
49.7025 (tx? - 15) . . 55.830784
34.0239

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.341883 : .0051679 And consequently x=10.6-,0051679=10.5948321, very nearly.

EXAMPLES FOR PRACTICE.

1. Given x3 + 10x2 +5r=2600, to find a near approximate value of x.

Ans. ==11.00673 2. Given 2x4 – 16x3 + 40x3 30x1=0, to find a near "value of x.

Aps. =1.284724 3 Given x5+2x4+3x3+4x' +5x=54321, to find the value of x.

Ans. 8414455 4. Given 703 +4x3 +720x3 - 10x=28, to find the value of x.

Ans. 4.510661 5. Given ✓ 144x2 - 6x2 +20): +v 196x2 -(x2 +24)? =114, to find the value of x.

Aps. 7.123883

OF EXPONENTIAL EQUATIONS.

An exponential quantity is that which is to be raised to some unknown power, or which has a 'variable quantity for its index ; as

af, af, zt, or 2, &c. And an exponential equation is that which is formed between any expression of this kind and some other quadtity, whose value is known; as

S. fi
a=b, paa, &C

Where it is to be observed, that the first of these

equa. tions, when converted into logarithms, is the same as

log. 6 a log. a=b, or x= ; and the second equation w*=a,

log. a is the same as x log. x=log. A.

In the latter of which cases, the value of the unknown quantity x may be determined, to any degree of exactness, by the method of double position, as follows :

RULE. J, B. Vousworthy

Find, by trial, as in the rule before laid down, twe numbers as near the number sought as possible, and substitute them in the given equation.

x log. x=

log. a, instead of the unknown quantity, noting the results obtained from each.

Then, as the difference of these results is to thé difference of the two assumed numbers, so is the difference between the true result, given in the question, and either of the former, to the correction of the number belonging to the result used; which correction being added to that number, when it is too little, or subtracted from it, when it is too great, will give the root required, nearly.

And, if the number, thus determined, and the nearest of the two former, or any other that appears

to be nearer, be taken as the assumed roots and the operation be repeated as before, a new value of the unknown quantity will be obtained still more correct than the first; and so on, proceeding in this manner, as far as may be thought necessary

EXAMPLES.

1. Given xt=100, to find an approximate value of x. Here, by the above formula, we have

z log x=log. 100=2. And since x is readily found, by a few trials, to be

nearly in the middle between 3 and 4, but rather nearer the latter than the former, lot 3.5 and 3.6 be taken for the two assumed numbers.

Then log. 3.55.5440680, which, being multiplied by 3.5, gives 1.904238=first result ;

And log. 3.63.5563025, which, being multiplied by 3.6, gives 2.002689 for the second result.

Whence 2.002689 3.6 . . 2.002689 .904238 3.5

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1.099451

.002689 : 00273 For the first correction ; which, taken from 3.6 leaves

3.59727, nearly. And as this value is found, by trial, to be rather too small, let 3.59727 and 3.59728 be taken as the two assumed numbers.

Then log. 3.597275.5559731, which being multiplied by 3.59727, gives 1.9999854= first result.

And log. 3.59728=.5559743, which, being multiplied by 3.51728, gives 1.9999953=second result.

Whence 1.9999953. . 3.59728 2. 1.9999854 3.59727 1.9999953

.

:

.0000099 .00001 .0000047 : 00000474747 For the second correction; which, added to 3.59728, gives x=3.59728474747, extremely near the truth. 2. Given 2*=2000, to find an approximate value of x.

Ans. x=4.82782263 3. Given (6x) =96, to find the approximate value

Ans. *1.8826432 4. Given r=123456789, to find the value of x.

Ans. 86400268

of x.

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5. Given 24-2=(2x^2+)*, to find the value of x

Ans. x=1.747933

07. THE

BINOMIAL THEOREM.

.

m

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m

m

m-n

m

AQ+

N

TAE binomial theorem is a general algebraical ex. pression, or formula, by which any power, or root of a given quantity, consisting of two terms, is expanded into a series ; the form of which, as it was first proposed by Newton, being as follows : )Pn

2n

2n
mainya?, &c.]
3n
Or

- 2n
(PtPQ)ñ=p7+

BQ+ cet 2n

3n - 3n

4n Where p is the first term of the binomial, Q the second term divided by the first, the index of the power, or

root, and A, B, C, &c. the terms immediately preceding - those in which they are first found, including their sigus +os.

Which theorem may be readily applied to any particular case, by substituting the numbers, or letters, in the given example, for P, Q, m, and n, in either of the above formulæ, and then finding the result according to the rule (1).

m

DQ, &c.

m

n

(l) This celebrated theorem, which is of the most extensive

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EXAMPLES.

1. It is required to convert (12-+x)+ into an infinite

series.

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ese in algebra, and various other branches of analysis, may be otherwise e pressed as follows:

m man m - 2n (a+x)ñ=añ[1+ +

)2 + 2n

2n 3n

m

m

m m-LX.

n

n

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-)3] &c.

c.

a ta It may

here also be observed, that if m be made to represent any whole, or fractional number, whether positive or negative,

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