286 A L G E 
the numerators together for a new numerator, and the de¬ 
nominators for a new denominator. Let j and — be the 
rrr , fl C OC . ffl " "C 
two tractions, then -x-——. For it -r— x > and -zry, 
b d bd a b a 
by multiplying the equal quantities - and x by b, az=.bx\ 
in the fame manner c—dy, and by the fame axiom ac—bdxy ; 
dividing thefe equal quantities, ac and bdxy, by bd, we 
ltave 
bd 
— xy- 
a c 
To divide a fraction by any quantity, multiply the de¬ 
nominator by that quantity, and retain the numerator. The 
fraCtion - divided by c is—. Becaufe and acth 
* 1 be b be 
P ar 
‘of this is —; the quantity to be divided being acth 
be 
y 
b d 
or <z 2 4 zab-\-P the fquare, or fecond power. 
£Z-f b 
d s -\- 2 d 2 b-\-aP 
4 d 2 b-\- 2 ab 2 -\-P 
i<4C} 3 or « 3 4 zdb-Y^aP^P the third power. 
B R 
a4C) 3 or 
A. 
* 
« 3 4-3a 2 i-f 2(ib*-\-b 3 the third power, 
a-\-b 
^■*43 db-\- 3 db"-\-aP 
4 db 3 a~b~-\ ^2p-\-P 
part of what it was before, and the divifor the fame. 
Cor. The refultis the fame whether the denominator is 
multiplied by the quantity or the numerator divided by it. 
CLC 
Let the fraction be —; if the denominator be multiplied 
bye, it becomes or the quantity which arifes 
bde bd 
from dividing the numerator by c. 
To divide one fraction by another, invert the numerator 
and denominator of the divifor, and proceed as in multi¬ 
plication. Let y and be the two fractions, then T 4 *w 
a d ad b d a c 1 , d 
y-—~. Proof. Let x, and ! then a—bx, 
b e be b d a d bdx 
and c—dy, alfo ad—bdx, and bc—bdy \ therefore, — —— 
v a c be bdy 
or cd-\-ya''b-\- 6 dd-\-yaP-\-P the fourth power. 
If b be negative, or the quantity to be involved a — b, 
whenever an odd power o f b en ters, the fign of the term 
mull be negative; hence a — b^ — ad — ya 3 b-\-Ga"P —4 ab 3 
+b\ 
Evolution, or the extraction of roots, is the method of 
determining a quantity, which, raifed to a propofed pow¬ 
er, will produce a given quantity. 
Since the nth power of a m is a mn , the nth root of a mn 
mu ft be a m ; i. e. to ex trad! any root of a (imple quantity, 
we muft divide the index of the propofed quantity by the 
index of the root required. 
Cor. The fquare, cube, fourth, nth, root of a, are pro- 
X x x — 
perly reprefented by a-, a 3 , a*, a“ ; and in like manner the 
lame roots of any compound quantity ; a 2 44 are proper- 
-x --ft- •- 1 -,x 
ly reprefented by dr 44"> d'-j-.x-j*, o 2 4x 2 f''; 
alfo the nth root of the mth power of a, is exprelfed by 
a n , of a 4-v, by &c. and the fquare, cube, fourth, 
x _. • . _i J, 1_ 
a ', by a 2 ,a 3 , a +, Sec. and of 
& c. root of - or 
a 
a 2 4- v 
— or£'4* l S bya'+x 2 . 
On Involution and Evolution. 
If a quantity be continually multiplied by itfelf, it is faid 
to be involved or-raifed ; and the power to which it is raif¬ 
ed is exprelled by the number of times the quantity has 
been employed in the multiplication. ThusaX«> or a", is 
called the fecond power of a ; eyayq, or a 3 , the third 
power; a", the nth power. 
If the quantity to be involved be negative, the figns of 
the even powers will be politive, and the figns of the odd 
powers negative. For— ay—a-j^a"-, —— a — az= — a?, 
Sec. 
A limple quantity is raifed to any power by multiplying 
the index of every factor in the quantity by the exponent 
of the power, and prefixing the proper fign determined by 
the lafit article. Thus a m , raifed to the nth power, is a mn . 
Becaufe a m yd n ya m — to n factors, by the rule for mill- 
- ■ — . >j 
tiplication, is a™"; ab] —abyabyaby Sec. to n factors, 
or ayaya.... to n factors, ylybyb.... to n factors, — 
a n yb n ; a"Pc raifed to the fifth power is a u ‘b' s c b . Alfo 
__ a " 1 raifed to the nth power is d za mn , where the pofitive or 
negative fign is to be prefixed according as n is even or 
odd. 
If the quantity to be involved be a fraCtion, both the 
numerator and denominator muft be raifed to the propofed 
power. 
If the quantity propofed be a compound one, the invo¬ 
lution may either he reprefented by the proper index, or it 
may actually take place. 
Let a-\-b be the quantity to be raifed to any power. 
<24^ 
a-\-b 
a 2 -\-ab 
-\-ab-\-P 
Gd-h-V) 3 , <2-4.4 4> 
Sec. Llere we fuppofe the rule for the multiplication 
tion of the powers of the fame quantity to extend to roots : 
1 ' x 
and fince, according to this rule, a 2 yd 2 —a', the fquare 
root of a', or a, is properly reprefented by d 2 . Ao-ain, 
_.1 _x _x _ _x 0 
a 3 ya 3 ya 3 '=a ‘, therefore <z 3 reprefents the 
cube root of a 1 ; and the fame reafoning extends to the 
other examples. 
If the root to be extracted be reprefented by an odd 
number, the fign of the root will be the fame with the 
fign of the propofed quantity. If the root to be extract¬ 
ed be reprefented by an even number, and the quantity 
propofed pofitive,' the root may be either pofitive or ne¬ 
gative. Becaufe either a politive or negative quantity, 
raifed to fuch a power, is pofitive. 
If the root propofed to be extracted be reprefented by 
an even number, and the fign of the propofed quantity be 
negative, the root cannot be extracted ; becaufe no quan¬ 
tity raifed to an even power can produce a negative refult. 
Such roots are called impojfible. 
Any root of a produCt may be found by taking that 
root of each faCtor, and multiplying the roots, fo taken, 
together. Thus ao^—a" yb ' 1 ; becaufe a~yid‘ raifed to 
the »th power is ab. 
Any root of a fraction may be found by taking that root 
of both the numerator and denominator. Thus, the cube 
„a 2 . a\ 2 ad a± 
rootofor a 3 X^ ; —“p- 
The method of extracting the root of a compound 
quantity will be underftood by attending to the involution. 
The fquare root of a 2 -\-2ab-\-b‘ is known to be a-\-b. In 
order then to eftablifh a general rule for the extraction of 
the fquare root, obferve in what manner a and b may be 
determined from a 2 -\-2ab-\-P. 
Having arranged the terms ac¬ 
cording to the dimenfions of one 
letter, (a,) the fquare root of the 
firft term a 2 is a, the firft faCtor 
in the root; fubtraCt its fquare 
from the whole quantity, and 
bring down the remainder 2ab-\-b 2 ; divide zab hy 2a, and 
the refult is b, the other factor in the root; then multiply 
the fuin of twice the firft faCtor and the fecond (za-\-b). 
b Y 
a 2 -\-2ab-\-P(a-\-b 
a 2 
2a-\-b)zab~\-P 
2ab-\-P 
