algebra. 
end of it, which number is called the index 
or expo nent. T hus, a- j-b\ 2 denotes the 
same as a -j - b x a - j-6 or second power , 
or square, of tr -4— 6 considered as one quan- 
which is 2. The 3ign y/ with a figure 
over it is used to express the cubic or bi- 
quadratic root, &c. of any quantity; thus 
V 64 represents the cube root of 64, or 4, 
tity; and u-f-A]’ denotes the same as a -f 6 because 4 X 4 X 4 is 64; and ab-\-cd 
X «-j-i x «-|-4, or the third power, or 
cube, of a b. In expressing the powers 
of quantities represented by single letters, 
the line over the top is usually omitted; 
thus, a 2 is the same as a, a or a x a, and 6 ! 
the same as 6 b b or b x b x b, and a 2 b 3 , the 
same as aa X b b b or a x a x b X b X b. 
The full point . and the word into, are some- 
times used instead of x as the sign of mul- 
tiplication. Thus, a-\-b . a-\-c, and a -{-IT 
into «-j- c, signify the same thing as a-\-b 
X « -f- c, or the product of « -j- b by «-J- c . 
The sign ~ is the sign of division, as it de- 
the cube root of a 6-J-cr/. In like manner 
V 16 denotes the biquadratic root of 16, 
or 2, becau se 2 X 2 X 2 X 2 is 16, and 
’s/ ab c d denotes the biquadratic root 
oi'ahS-cd] and so of others, Quantities 
thus expressed are called radical quantities, 
or surds ; of which those, consisting of one. 
term only, as \J a and are called 
simple surds; and those consisting of seve- 
ral terms or n umbers, as \/ a 2 — b l and 
notes that the quantity preceding it is to be 
divided by the succeeding quantity. Thus, 
c-^-b signifies that c is to be divided by b; 
and a -j- b — a c, that a -j- b is to be di- 
vided by a -)- c. The mark ) is sometimes 
used as a note of division ; thus a -j- b) a, b, 
denotes that ab is to be divided by a -j- b. 
But the division of algebraic quantities is 
most commonly expressed bv placing the 
divisor under the dividend with a line be- 
tween them, like a vulgar fraction. Thus, 
c . ... 
g represents the quantity arising by - di- 
viding c by b, or the quotient, and 
represents the quotient -of a h 
divided by c-(- c. Quantities thus express- 
ed are called algebraic fractions. 
The sign \/ expresses the square root 
of any quantity to which it is prefixed ; thus 
aJ 25 signifies the square root of 25, or 5, 
because 5x5 is 25 ; and a/ a b denotes 
the square root of ab-, and \/ — 
denotes the square root of or of the 
quantity arising from the division of a b 4- 
Z. I L „ 
; but n ^ ^ which has 
be by <2; d 
the se- „ _ 
parating line drawn under \/ , signifies 
that the square root of a b -j- b c is to be 
first taken, and afterwards divided by 
d ; so that if a were 2, b, 6, c, 4, and d, 9, 
'±/ ± b + bc would be 
d 
^ or 
9 9’ 
but 
\/ “/' .. "j - -f would be \/ ^ or y' 4 , 
d 
t/V^-b 
+ * c are denominated compound 
surds. Another commodious method of 
expressing radicaliquantities is that which 
denotes the root by a vulgar fraction, 
placed at the end of a line drawn over 
the quantity given. In this notation, 
the square root is expressed by j, the 
cube root by 'the biquadratic root by 
& c. . Thusj?i expresses the same quan- 
tity with a/ a, i. e. the square root of «, 
and a 2 -j-a"ir ]3 the same as y*+ub, i. e . 
the cube root of a 2 -j- a b • and < 7 | : denotes 
the cube root of the square of a, or the 
square of the cube root of a; and 
the seventh power of the biquadratic root 
of a -J- z; and so of others; a 2 )? a 
a 5 ] 2 is a, &c. When the root of a quantity 
represented by a simple letter is to be ex- 
pressed, the line over it may be omitted • 
so that a 2 signifies the same as"o)l, and bi 
the same as T? or Quantities that 
have no radical sign [ v r ) or index an- 
nexed to them, are called rational quanti- 
ties. The sign =, called the sign of equa- 
lity, signifies that the quantities between 
which it occurs are equal. Thus 2 -f- 3 = 5 
shews that 2 plus 3 is equal to 5; and 
x x=a^—b shews that .r is equal to the dif- 
ference of n and b. The mark :: signifies 
that the quantities between which it stands 
are proportional. As a : b :: c:d denotes 
that a is in the same proportion to b as c is 
to d ; or that if a be twice, three, or four 
times, See. as great as b, c will be twice, thrice 
or four times, &c. as great as d. When any 
quantity is to be taken more than once, the 
number, which shews how many times it is to 
