52 
ALGEBRA. 
tients in their room, and you shall have a frac- 
tion equivalent to the given fraction expressed 
in the least terms. Thus, 
C 25bc \ 75aic 3a 156 aa -|— 1 56aA 
2 5bc ' \2obcx 5v’ 572aa — 572aA 
3 a -f- 3b a 2 — b 2 _ a -f- b _ 
Hi? 
11 a 
a 5 - b 2 a 
a 2 -j- 2 ab b 2 
a 2 -j -b 1 
— 2 ab -}- A 2 a — 
a 2 — ba a 4 — A 1 
a -}- b * a ’ — a ’A" 
b ’ 
When unit is the greatest common measure 
of the numbers and quantities, then the frac- 
3 ab 
tion is already in its lowest terms. Thus, — - 
cannot be reduced lower. 
And numbers, whose greatest common mea- 
sure is unit, are said to be prime to one another. 
If it is required to reduce a given fraction to 
a fraction equal to it that shall have a given de- 
nominator, you must multiply the numerator 
by the given denominator, and divide the pro- 
duct by the former denominator, the quotient, 
set over the given denominator, is the traction 
required. Thus — being given, and it being 
b 
required to reduce it to an equal fraction whose 
denominator shall be c ; find the quotient of ac 
divided by b, and it shall be the numerator of 
the fraction required. 
Of the Involution of Quantities. 
The products arising from the continual mul- 
tiplication of the same quantity, were before 
called the powers of that quantity. Thus a, a\ 
T, a 4 , See. are the powers of a - and ab, a A b 2 , 
a P, a 4 A 4 , & c. are the powers of al>. And the 
rule given for the multiplication of powers of 
the same quantity was, to “ Add the exponents, 
and make their sum the exponent of the pro- 
duct.” Thus a 4 X a’ ~ a’ ; and a l P X a< b l = 
TA\ In the above place you have the rule for 
dividing powers of the same quantity, which is, 
“ To subtract the exponents, and make the dif- 
ference the exponent of the quotient.” 
T 
Thus, — r = P - 4 = a 2 ; and — = 
T - 4 P - 1 = ab 2 . 
If you divide a lesser power by a greater, the 
exponent of the quotient must, by this rule, be 
— a 4 ~ 6 = a ~ 2 . Eut 
i \ ~ 6 4- 4 
— a w * (or — ) ; also a ~ 6 X « 4 = a 
— a~ 2 (or ; and a — 3 X a' = a 0 — 1 . 
a ' 
And, in general, any positive power of a mul- 
tiplied by a negative power of a of an equal 
exponent, gives unit for the product v for the 
positive and negative exponents destroy each 
other, and the product gives a 0 , which is equal 
to unit. 
-• + 
Likewise - — r = 
and ■— 
— a\ Eut also, 
X a- 
— — — ; therefore — — s = a 3 
And, in general, “ any quantity placed in the 
denominator of a fraction, may be transposed 
to the numerator, if the sign of its exponent be 
changed.” Thus — p =z a — 3 , and — == T. 
The quantity a m expresses any power of a in 
general ; the exponent (»z) being undetermined; 
1 
and a ~ m expresses — , or a negative power 
of a of an equal exponent : and a m X a~ m — 
a m — m — a 0 — l is their product, a" expresses 
m -j- n . 
any other power of a • a m X att — a ls 
the product of the powers a m and a ”, and 
a m — n ; s their quotient. 
To raise any simple quantity to its second, 
third, or fourth power, is to add its exponent 
twice, thrice, or four times to itself ; therefore 
the second power of any quantity is had by 
doubling its exponent, and the third by treb- 
i ling its exponent ; and, in general, the power 
! expressed by m of any quantity is had by mul- 
tiplying the exponent by m, as is obvious from 
the multiplication of powers. Thus the second 
a b — Root. 
X a 
a 2 -J- ab 
-}- ab -j- b 2 
negative. 
a 4 1 
Thus, —r 
1 
and hence — - is expressed also by 
T a 1 - a 
a 2 with a negative exponent, or a — 2 . 
a . . . 
It is also obvious, that — — « ~ = « ; 
but — — 1 , and therefore a 0 
a 
1 a 0 
same manner, — ~ — =■ < 
a a 
1 «° 
1 . Aftqr the 
1 , 
« - 3 ; so that the quantities 
JL -L-, &c. may be expressed thus, a 1 , a 0 . 
„ - \ a — 2 , a — 3 , a- 4 , & c. Those are called 
the negative powers of a , which have negative 
exponents ; but they are at the same time posi- 
- 1 
tive powers of — , or a 
a 
Negative powers (as well as positive) are 
multiplied by adding, and divided by subtract- 
ing, their exponents. Thus the product of a — 2 
(or -ig) multiplied by a 
power or square of a is a — a 7 ; its third 
3 x i 
power or cube is n re a 3 ; and the ruth 
„ . » X 1 . , , 
power of a is a — a m . Also, the square 
o 3 X 
of a 4 is a" ' — a* ; the cube of a 4 is a 
4 ^ T7Z 
— a n ; and the »zth power of a 4 is a 
The square of abc is a 1 b 2 A, the cube is a 2 A 3 c\ 
the wth power a m b m c m . 
The raising of quantities to any power in 
called Involution ; and any simple quantity is 
involved by multiplying the exponent by that 
of the power required, as in the preceding ex- 
amples. 
The co-efficient must also be raised to the 
same power by a continual multiplication of 
itself by itself, as often as unit is contained in 
the exponent of the power required. Thus the 
cube of 3 ab is 3 X H X 3 X a b' — 27 a l P. 
As to the signs, When the quantity to he ih>- 
volved is positive, it is obvious that all its 
powers must be positive. And when the quan- 
tity to be involved is negative, yet all its powers 
whose exponents are even numbers must be 
positive ; for any number of multiplications of 
a negative, if the number is even, gives a posi- 
tive ; since — X — = 4~> therefore — x — 
X — X — = + X + = + 5 and — x — 
X— X— X — X — -- 4-X+X + — 
4 -1 ’ 
The power then only can be negative when 
its exponent is an odd number, though the 
quantity to be involved be negative. The powers 
of — a are — a, -{- a 1 , — a 3 , -j- a 4 , — a’,^Scc. 
Those whose exponents are 2 , 4, 6, See. are posi- 
tive ; but those whose exponents are 1, 3, 5, See. 
are negative. 
The involution of compound quantities is a 
more difficult operation. The powers of any 
binomial a 4” a are found by a continual multi- 
plication of it by itself, as follows. 
a 2 -f- 2 ab -j - b 2 — the Square, or 2d Power. 
X a 4" A 
a 3 -j- 2 a 2 b — ab 2 
-j- a 2 b — 2 ab 2 -j- A 1 
T- 
X a -\ 
. 3 TA -j- 3 a A 2 4 - A 5 = Cube, or 3d Power. 
-A 
P _L 3 TA 4 - 3a 2 P 4- 
ab' 
-f TA-j-3TA 2 -j- 
3aP -}- A 4 
“ 4 H 
- 4« 5 A 4 - Ga 2 P -}- 4a A 3 -|- A 4 = Biquadrate, 
X a H 
hA 
T -j- 4TA 4- 6a A 2 - 
- 4a 2 A 3 -}“ "A 4 
TA-}- 4 TA 2 - 
- 6 a 2 A 3 -j- 4aA 4 4 - P 
X a 
T -j- 5a'b 4- IOTA 2 4- 10a 2 A 3 4- 5ap 4- A = 5th Power. 
5Pb -4- IOTA 2 -f- IOTA 5 4- 5 a % 4 4 - aP 
— - TA -j - * 5a 4 b 2 4~ 10 a A 3 4" IOTA 4 4" SaP 
T 4- 6 TA 4- 15TA 2 -f 20 TA 3 -J- 15TA 4 4 - Gap 4~ P = 6 th Power, &c. 
If the powers of n — A are required, they 
will be found the same as the preceding, 
only the terms in which the exponent of 
A is an odd number will be found negative; 
« because an odd number of multiplications of 
a negative produces a negative.” Thus, the 
cube of a — A will be found to be a 3 — 3a 2 b 
4- 3ab 2 — P ; where the 2d and 4th terms are 
negative, the exponent of A being an odd num- 
ber in these terms. In general, “ the terms of 
( ■■ I A is a -z -3 \ any power of a — b are positive and negative 
J a _ I by turns.” 
It is to be observed, that “ in the first term 
of any power of a b, the quantity a has the 
exponent of the power required ; that in the 
following terms, the exponents of a decrease 
gradually by the same difference, (viz. unit) ; 
and that in the last terms it is never found. 
The powers of A are in the contrary order ; it is. 
not found in the first term, but its exponent in 
the second term is unit, in the- 3d term its expo- 
nent is 2 ; and thus its exponent increases, till in. 
the last term it becomes equal to the exponent 
of the power required.” 
