ALGEBRA. 
To divide a fraction, by any quantity, multi- 
ply the denominator by that quantity , and retain 
the numerator . 
Thefraction % divided by c, is-— . Because 
b be 
to the denomirfator, and the contrary, by 
changing the sign of its index. Thus, 
a m X a n a m . a m a 7 ' 1 X 
■ " — i and - — : — — ... 9 
be __ bta~" a n bP bp 
ON INVOLUTION AND EVOLUTION. 
r—~r, and a c* wart of this is—-; the 
o be o c 
quantity to be divided being a c th part of 
■what it was before, and the d visor the same. 
The result is the same, whether the deno- 
minator is multiplied by the quantity, or the 
numerator divided by it. 
Let the fraction be ; ; if the denominator 
bd 
a c a 
he multiplied by c, it becomes or — ; 
the quantity which arises from the division 
of the numerator by c. 
To divide one fraction by another, invert the 
numerator and denominator of the divisor, and 
proceed as in multiplication. 
Let ~ and % be the two fractions, then rfr~i 
b d b a 
a d ad 
b ^ c be 
For if | = x, and ~ = i I, then avzcbx, 
and c ~ dy ; also, ad “ b d x, and b c czz 
ad bdx cc a.c 
bdy, therefore ^7=4^ = - ~b~r 
The rule for multiplying the powers of the 
same quantity will hold when one or both 
of the indices are negative. 
Thus, a m X a- n = a m - n ; for a m X a~" = 
1 a m . 1 
a m x — a m ~ n ; in the same manner, 
a H a 1 
X 3 1 , 
*5 X X S =~ — “= * • 
Again, a~ m X a~ n — a~'“ +n \ because a~ m 
^ _J —• '"+» 
If m — n, a m X a~ m — a’"~ m = a’; also. 
a m X “~ m — -7 = 1 ; therefore a° — 1 ; 
according to the notation adopted. 
The rule for dividing any power of a quan- 
tity by any other power of the same quan- 
tity holds, whether those powers are positive 
or negative. 
Thus, a m -t- a~ n = a™ -f- *-= <t m X a” = 
„m+n. 
Again, a-”-r«-’ = ~i = ;= a ^ 
Hence it appears, that a quantity may be 
transferred from the numerator of a fraction 
Involution. If a quantity /be continually 
multiplied by itself, it is said to be involved, 
or raised ; and the power to which it is ra ; s- 
ed is expressed by the number of times the 
quantity has been employed in the multipli- 
cation. 
Thus, a X «, or a 2 , is called the second 
power of a ; a X a X «> or a 3 , the third 
power; a X a ( n ), or a”, the n th 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. 
For — a X — a — a 2 -, — a X — a X — a 
= — a 3 , &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, o m raised to the n th power is a mn . Be- 
cause a m X a m X a m . . . to n factors, by the rule 
of multiplication, is a mn ; also; a~D ' — a b X 
ab X ab X &c. to n factors, or a X a x a 
.... to re factors x b X b X b ton 
factors = a" X b n ; and a 2 6’- c raised to the 
fifth power is a 10 b li c s . Also, — a m raised to 
the re th power is ^a m "; where the positive or 
negative sign is to be prefixed, according as 
re is an even or odd number. 
1 If the quantity to be involved be a frac- 
tion, both the numerator and denominator 
must be raised to the proposed power. 
If the quantity proposed he a compound 
one, the involution may either be represent- 
ed by the proper index, or it may actually 
take place. 
Let a -j- b be the quantity to be raised to 
any power. 
a + b 
a b 
a 2 -(- a b 
+ ab + b 2 
a x o\ or a 2 2 a b -j- b 2 the sq. or 2^ power 
a -}~ b 
a 1 — 2 a 2 b •— a b 2 
-j- a 7 b 2 ab 7 — 1— 
a — |— i/l or a J -f- ‘J u 2 b -j- 3 a b 2 -j- b 3 the 3 ^ p. 
a -j-A 
a* -J- 3 a* b 3 b 7 -j- a A 3 
' _j_ b -j- 3 a 7 b 2 -j- 3 a 6 3 -|~j4 
a -j- b 1 1 or ai-f-4a 3 A-j-6a-A--|-4ai J -j-6t 
the fourth power. 
