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shew that the remainders resulting from the conversion of a fraction into a decimal will recur in a certain order.

3

Will if converted into a decimal, terminate? And if not what

19'

is the greatest number of digits which the repeating period could contain?

5. Prove the rule for finding the Least Common Multiple of two algebraical expressions, and extend it to three.

Find the L. C. M. of 1−x+x2, 1+x, and 1-x2.

6. State the Rule for pointing in extracting the square root of a whole number, and also of a decimal.

How many hurdles, each 21 yards long, will enclose a square field of 10 acres?

C.

a

a

ma

1. What does the expression signify? Shew that b mb' State and prove the rules for the Multiplication and Division of Algebraical Fractions.

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2. Which is the greater, 1414235, or the square root of 2? Find

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4. Shew that [{(am)"}]=am2o, and find the value of

(1 + x2 + ∞ + œ13)2 (1 − ∞ 13) 3 {1 — x (x − 2)}.

5. Prove the rule for finding the Greatest Common Measure of three Algebraical Expressions.

Find the G. C. M. and also the L. C. M. of

6 (a3-b3) (a - b)3, 9 (a1-b1) (a-b), and 12 (a-3)3.

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6. Investigate a Rule for extracting the cube root of a Compound Algebraical Expression.

Extract the cube root of

(a+1)o1μ3 — 6cao (a+1)1x2+12c2a2o (a+1)2x − 8c3å3r.

D.

1. Prove the rules for the addition and subtraction of Algebraical

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and reduce

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2x3+ (2α-9) x2 - (9a+6) x + 27 and 2x2-13x+18,

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3. Explain the method of pointing in extracting the cube root of a whole number, and also of a decimal.

Find the cube root of 000102503232; and find within an inch the length of the side of a square field the area of which is 2 acres.

4. Shew that every common multiple of (a) and (b) is a multiple of their least common multiple.

Find the L. C. M. of

2x2-xy-3y2 and 2x2-5xy+3y2.

5. Simplify the following expressions:

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6. Multiply the cube of x3+a3 by the square of x+a; and shew

that the product

= (x+a)2 + 2x31a3 (x+a)+3x*a* (x2+a3)*(x*+
$+a3).

SIMPLE EQUATIONS.

99. Such an expression as (x+1)2 = x2+2x+1, where one of the quantities, between which the sign of equality is placed, results from performing the operations indicated in the other, is called an " IDENTITY." An expression of this kind merely asserts that an algebraical quantity in one form is equivalent to the same quantity in another form; and it will clearly hold true for all values of x.

Such an expression as (x+1)2=16 or x2+2x+1=16, where the quantity on one side of the sign of equality cannot be made to result from the other by performing the operations indicated in either, is called AN EQUATION; and in such a case the equality will only exist for certain values of x. We may define an equation as follows:

Def. When two quantities, differently expressed, and one of which does not result from the other, are connected together by placing the sign=between them, the whole expression forms AN EQUATION. Thus x=ab is an equation which shows that x is to be equal to the product of a and b.

100. If an equation, which has been cleared of fractions and surds, contain the first power only of an unknown quantity, it is called a SIMPLE EQUATION, or an equation of the first degree, or an equation of one dimension. If it contain the square only of the unknown quantity, either with or without the first power, it is called a QUADRATIC EQUATION, or an equation of the second degree, or an equation of two dimensions. Thus x—a=b is a simple equation; and x2+16=25, x2-x=5, are quadratic equations.

Generally, if the index of the highest power of the unknown quantity be n, then the equation is said to be of the nth degree, or of n dimensions.

101. To solve an equation is to find that value or those values of the unknown quantity or quantities contained in it, which will satisfy the equation; that is, which will, when substituted for the unknown quantity or quantities, make the first side of the equation equal to the second.

The value or values of the unknown quantity or quantities which satisfy the equation, are called THE ROOT OR ROOTS of the Equation.

102. The solution of equations depends on the following Axioms, or self-evident Propositions :

Ax. 1. If equal quantities be added to or subtracted from equal quantities, the sums or remainders will be equal.

Ax. 2. If equal quantities be multiplied or divided by equal quantities, the products or quotients will be equal.

Ax. 3. If equal quantities be raised to the same power, or have the same root extracted, the results will be equal.

103. The Rules absolutely necessary for the solution of simple equations which contain one unknown quantity, may be reduced to the four following:

RULE 1. Any quantity may be transferred from one side of an equation to the other, by changing its sign.

This rule is founded on Ax. 1.

Let x+a=b.

Subtract a from each side of the equation. Then

x+a-a=b-a,

but a-a=0;

.. x=b-a,

where a is on the right-hand side, and has changed its sign.

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where a has changed sides, and also its sign.

Again, let 4x-20=3x+5.

Add 20 to each side of the equation, then

4x-20+20=3x+5+20,

or 4x=3x+5+20, (where the 20 has changed sides and sign), =3x+25.

Subtract 3x from each side,

Then 4x-3x=3x+25-3x,

=25, (where 3x has changed sides and sign), = 25.

or x=

COR. 1. From this Rule it is evident that if the same quantity be on each side of an equation with the same sign, it may be left out of the equation.

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x-a-b-a-c is the same as x-b-c.

All the signs of an equation may be changed from + to to +, without altering the value of the unknown quantity

Let x-c=b-a, (1),

then x-b-a+c.

Now if all the signs of (1) be altered,

−x + c = −b+a,

then c+b-a=x, as before.

RULE 2. If the unknown quantity be multiplied by any quantity, the latter may be taken away by dividing all the terms of the equation by it.

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RULE 3. If the unknown quantity be divided by any quantity, the latter may be taken away, by multiplying all the terms of the equation by it if there be more than one such fraction, multiply all the terms of the equation by the least common multiple of the denominators.

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L. C. M. of the den"= 6; therefore multiply each term by 6.

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