ALGEBRA. 
If b be negative, or the quantity to be in- 
volved be a — b, wherever an odd power of 
A enters, the sign of the term must be nega- 
tive. 
"i) 4 — n't — 4 a.3 b -j- 6 a 2 b 2 
, "i l * 
1 = — .ora * xi 
Hence, a — 
•*— 4 a bz 44. 
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 ra th power of a m is a mn , the n th 
root of a mn must be a m ; 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 ex- 
presses the root to be extracted, that root 
must be represented according to the nota- 
tion already pointed out. 
Thus, the square, cube, fourth, n th root of 
a 2 -j- x 2 , are respectively represented by 
~,ora 3 X b 3 ■ and- 
** , " J bn 
To extract the square root of a compound 
quantity. 
a 2 -j- 2 a b -f- b 2 {a -f 4 
a 2 
2 a -}- 4) 2 a b -f- 4 2 
2 a -4 -j- 4 2 
a 2 -\-X 2 \ 2 ' a 2 -J- x 
the same roots of 
represented bya 2 -j-a 
a-' -j-rTj n- 
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 nega- 
tive 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 quan- 
tity, raised to an even power, can produce 
a negative result. Such roots are called 
impossible. 
Any aoot of a product may be found by 
taking that root of each factor, and multi- 
plying the roots, so taken, together. 
Thus, a bin — a" x A» ; because each of 
these quantities, raised to the n th power, is 
In a = 4, then an x a” = an ; and in the 
T S T-+. 
# * 
Since the square root of a 2 2 a b -j- A 2 is 
a+A whatever be the values of a and 4, we 
may obtain a general rule for the extraction 
of the square root, by observing in what 
manner a and 4 may be derived from a 2 4- 
2aA-j-A 2 . 
Having arranged the terms according to 
the dimensions of one letter, a, the square 
root of the first term, a 2 ,, is a, the first factor 
in the loot; subtract its square from the 
whole quantity, and bring down the remain- 
der 2 a A -j- A 2 ; divide 2 a A by 2 a, and the 
.result is 4, the other factor in the rpot; then 
multiply the sum of twice the first factor and 
the second (2 a -j- 4), by the second (4), 
and subtract this product (2a4-j-4 J ) from 
the remainder. If there be no more terms, 
consider a -j- 4 as a new' value of a; and its 
square, that is a 2 -f 2 a 4 -f A 2 , having, by 
the first part of the process, been subtracted 
from the proposed quantity, divide the re- 
mainder 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 a 2 4- 
2a4-j-4 2 -J-2ac-)-24c-(-c 2 . 
a 2 -f- 2 a 4 -f A 2 -j- 2 a c -j- 2 4 c -j- c\a -j- A-j- c 
2 a -j— 4)2 a b — |— A 2 
2 a A. A 2 
2 a — j— 2 4 — j— c)^ 1 
2ac-J-2Ac-|-c 2 
2ac-j-2Ac-|-c 2 
5 * W~ 
Ex. 2. To extract the square root of a 2 ax 
~ aX + l( Q — I' 
