denominator bdf. For

edb e

=

ad f a cb f bdƒ— b' b dƒ the numerator and deno

; and bdf minator of each fraction having been multiplied by the same quantity, viz. the product of the denominators of all the other fractions. When the denominators of the proposed fractions are not prime to each other, find their greatest common measure; multiply both the numerator and denominator of each fraction, by the denominators of all the rest, divided respectively by their greatest common measure; and the fractions will be reduced to a common denominator in lower terms than they would have been by proceeding according to the former rule.

Here a is considered as a fraction whose denomninator is unity.

If two fractions have a common denominator, their difference is found by taking the difference of the numerators and retaining the common denominator. Thus,

-

b C

The sign of bd is negative, because every part of the latter fraction is to be taken from the former.

ON THE MULTIPLICATION AND DIVISION OF

FRACTIONS.

To multiply a fraction by any quantity, multiply the numerator by that quantity and retain the denominator.

be divided be c times as great as before, and the divisor the same, the quotient must be c times as great.

The product of two fractions is found by multiplying the numerators together or a new numerator, and the denominators for a new denominator.

=

ac

be the two fractions; then d.

=y, by

For if = 1, and = bď

multiplying the equal quantities — and x, by b, abx; in the same manner, c=dy; therefore, a cbdxy; dividing these equal quantities, ac and bdxy, by bd, we have

a

= x, and {=y, then a = b x, and cdy; also, ad bdx, and be

с

=

bras and

Thus,

am a" be

=

ON INVOLUTION AND EVOLUTION.

INVOLUTION. If a quantity be continually multiplied by itself, it is said to be involved, or raised; and the power to which it is raised is expressed by the number of times the quantity has been employed in the multipli

cation.

Thus, a Xa, or a2, is called the second power of a; a × a × a, or a3, the third power; a X a.... (n), or a", the nth power. If the quantity to be involved be negative, the signs of the even powers will be positive, and the signs of the odd power negative.

-a

For a Xa — a2; — a X-ax=— a3, &c.

A simple quantity is raised to any power, by multiplying the index of every factor in the quantity by the exponent of the power, and prefixing the proper sign determined by the last article.

Thus, a raised to the nth power is am". Because a×a×a"... to n factors, by the rule of multiplication, is a""; also, arab x ab x ab x &c. to n factors, or a Xa Xa to n factors x b xbx b .... to n factors a X b"; and a2 b3 c raised to the fifth power is alo b15 c. Also, - a raised to the nth power is amn; where the positive or negative sign is to be prefixed, according as n is an even or odd number.

If the quantity to be involved be a fraction, both the numerator and denominator must be raised to the proposed power.

If the quantity proposed be a compound one, the involution may either be represented by the proper index, or it may actually take place.

Let ab be the quantity to be raised to any power.

axb2 or a2+2 a b + b2 the sq. or 2a power a+b

a 3 +2ab+ab2

+ a2b+2a b2 + b3

a + ¿13or a + 3 a2 b + 3 a b2 + b3 the 38 p.

a + b

a+ + 3 ab+3a2 b2 + a b3 + a3b+3a2 b2 + 3 a b3 + b¢ a + ¿11 or a4 + 4 a3 b + 6 a 2 b2 + 4 a b3 +b♦ the fourth power.

EVOLUTION, or the extraction of roots, is the method of determining a quantity which raised to a proposed power will produce a given quantity.

Since the nth power of am is am", the nth root of a must be a"; i. e. to extract any root of a single quantity, we must divide the index of that quantity by the index of the root required.

When the index of the quantity is not exactly divisible by the number which expresses the root to be extracted, that root must be represented according to the notation already pointed out.

I

Thus, the square, cube, fourth, nth root of a2x2, are respectively represented by a2+x3 31‚ «2+x13113‚, a2+x2, a2+x; or a2+x-', are

the same roots of

1

a2 + x 2

a2+2ab+b2 (a + b

Since the square root of a2 + 2 a b + b2 is a+b whatever be the values of a and b, we may obtain a general rule for the extraction of the square root, by observing in what manner a and b may be derived from a2+ 2ab+b2.

Having arranged the terms according to the dimensions of one letter, a, the square root of the first term, a2, is a, the first factor in the root; subtract its square from the whole quantity, and bring down the remainder 2 ab+b2; divide 2 ab by 2 a, and the result is b, the other factor in the root; then

multiply the sum of twice the first factor and the second (2 a+b), by the second (b), and subtract this product (2 a b + b) from

If the root to be extracted be expressed by an odd number, the sign of the root will be the same with the sign of the proposed quantity.

If the root to be extracted be expressed by an even number, and the quantity proposed be positive, the root may be either positive or negative. Because either a positive or negative quantity, raised to such a power, is positive.

If the root proposed to be extracted be expressed by an even number, and the sign of the proposed quantity be negative, the root cannot be extracted; because no quantity, raised to an even power, can produce a negative result. Such roots are called impossible.

Any oot of a product may be found by taking that root of each factor, and multiplying the roots, so taken, together.

consider ab as a new value of a; and its square, that is a2 + 2 a b + b2, having, by the first part of the process, been subtracted from the proposed quantity, divide the remainder by the double of this new value of a, for a new factor in the root; and for a new

subtrahend, multiply this factor by twice the sum of the former factors increased by this factor. The process must be repeated till the root, or the necessary approximation to the root, is obtained.

Ex. 1. To extract the square root of a2 + 2ab+b2+2ac + 2 b c + c2. a2+2ab+b2+2 ac+2bc+c2(a+b+c

The cube root of a3+3a2b+3a b2+b3 is a +b; and to obtain a + b from this compound quantity, arrange the terms as before, and the cube root of the first term, a3, is a, the first factor in the root; subtract its cube from the whole quantity, and divide the first term of the remainder by 3 a2, the result is b, the second factor in the root; then subtract 3a2b+3ab2+b3 from the remainder, and the whole cube of a+b has been subtracted. If any quantity be left, proceed with a+b as a new a, and divide the last remainder by 3.ab for a third factor in the root; and thus any number of factors may be obtained.

ON SIMPLE EQUATIONS.

If one quantity be equal to another, or to nothing, and this equality be expressed algebraically, it constitutes an equation. Thus, ■ — a — b — x is an equation, of which - a forms one side, and b - the other.

Hence, if the signs of all the terms on each side be changed, the two sides will still be equal.

Let xab-2r; by transposition, − b + 2 x = · -x +a; or a-x=2x-b.

If every term, on each side, be multiplied by the same quantity, the results will be equal.

An equation may be cleared of fractions, by multiplying every term, successively, by the denominators of those fractions, excepting those terms in which the denominators are found.

If each side of an equation be raised to the same power, the results will be equal.

Let r = 9; then x 9 × 981. Also, if the same root be extracted on both sides, the results will be equal. Let r81; then x = 9.

To find the value of an unknown quantity in a simple equation. ́ ́

Let the equation first be cleared of fractions, then transpose all the terms which involve the unknown quantity to one side of the equation, and the known quantities to the other; divide both sides by the co-efficient, or sum of the co-efficients, of the unknown quantity, and the value required is obtained.

by transp. 3x + x = 23+ 5

by division

x=

= 7.