SURD. 
^ a^ = Ay a‘' = \/ = ^ a% and 4 = 
V” 16 64= 256 = ^ 1024 = 
^ 4”. 
As surds may be considered as powers 
with fractional exponents, they are reduced 
to others Of the same value, that shall have 
the same radical sign, by reducing these 
fractional exponents to fractions having the 
same value and a common denominator. 
Thus ^ a = a”, and ^ a = om, and - = 
— , = i=~^ and therefore a/ a and v' 
jjTO m nm 
a, reduced tp the same radical sign, become 
and If you are to reduce 1/^3 
and 4^ 2 to the same denominator, consi- 
der ^ 3 as equal to Sij'and ^2 as equal 
1 
to 2^, whose indices, reduced to a common 
1 I i 
denominator, yo,u have 3^ = 3®, and 2^ = 
2‘ , and consequently, ^ 3 = 4^ 3’, = 
27, and 4^2='V^ 2^=4^4;so that the 
proposed surds >V o and 'v' 2, are reduced 
to other equal surds 27 and 4, hav- 
ing a common radical sign. 
Surds of the same rational quantity are 
multiplied by adding their exponents, 
and divided by subtracting them; thus, 
j .3 + 2 5 
^oX4^o = a2x«^=“‘’ =a‘ = 
and f-^=-^, = a^ ‘ =«— = 
V ' i/a ai 
2 m-\-n V a 
o“ = !y a'‘;v'aX^« = « mn'’~ir~ 
^ ' '' 
i/ % 
4' 3 
If the surds are of different rational quan- 
tities, as ^ and \/ b\ and have the same 
sign, multiply these rational quantities into 
one another, or divide them by one another, 
and set the common radical sien over tlieir 
product, or quotient. Thus, a‘ x \/ 
= ^^; ^2XAf5 = ^yiO;|5' = 
Vb-a 
^ h^a~^ 3/„. — A' aA — 
vK24 
24 
As 
:l^3. 
If surds have^ not the same radical sign, 
reduce them to such as shall have the 
same radical sign, and proceed as before; 
•V^ 3 X 4 = 2^ X 4l = 2‘x4® = 
^_2 ’ X 4^ = 4/s X 16 = ^T'g 8. 
^ ii — ^ — 
2 22 2^ 8 
^ 2 . 
If the surds have any rational coeffi- 
cients, their product or quotient must be 
prefixed ; thus, 2.^3x5.^6 = i0.^ 
18. The powers of surds are found as the 
powers of their quantities, by multiplying 
their exponents by the index of the power 
required ; thus the square of 2 is 2 s ^ 
= 2 s = /^ 4 ; the cube of 5 = 52 ^^ 
= 55 = .^ 123. Or you need only, in in- 
volving surds, raise the quantity under the 
radical sign to the power required, continu- 
ing the same radical sign ; unless the index 
of that power is equal to the name of the 
surd, or a multiple of it, and in that case 
the power of the surd becomes rational. 
Evolution is performed by dividing the 
fraction, which is the exponent of the surd, 
by the name of the root required. Thus, 
the square root of .^^*^is or ^ o'*. 
The surd a x = a !y x-, and, in 
like manner, if a power of any quantity 
of the same name with the surd divides 
the quantity under the radical sign without 
a remainder, as here o™ divides a™ x, and 
25 the square of 5, divides 75 the quantity 
under the sign in 75 without a remain- 
der ; then place tlie root of that power ra- 
tionally before the sign, and the quotient 
under the sign, and thus tlie surd will be 
reduced to a more simple expression. 
Thus, 75 = 5 \/ 3 ; ^ 48 = ^ 3x16 
= 4 3 ; 81 = ,^ 27 X 3 ='3 3. 
When surds are reduced to their least 
expressions, if they have the same irra- 
tional part, they are added or subtracted, 
by adding or subtracting their rational 
coefficients, and prefixing the sum or dif- 
ference to the common irrational part. 
Thus, ^75 + ^ 48 — 5^/3 3 
= 9 3 ; 4^ 81 -f 24 = 3 3 2 
3 = 3 y' S; 160 — y/ 5 4 = 5 y' 
6 — 3^6=2^76; •/ x b^ x 
= a^x-\-bj^x = a-\-bX >/ x. 
Compound surds are such as consist of 
two or more joined together ; the simple 
