If a is a rational number and n is a positive integer such that the nth root of a i.e. a1/n or is an irrational number, then a1/n is called a surd. It is also called a radical of order n, a is called the radicand.
i) is a surd. It can also be written as (3)1/2, Since 3 is a rational number and 2 is a positive integer and is an irrational number. So is a surd.
ii) is not a surd
Because 8 is a rational number and 3 is a +ve integer but =2 is not an irrational number. So is not a surd
iii) is not a surd because 2 + is not a rational number.
iv) Every surd is an irrational number but every irrational number is not a surd
LAWS OF SURDS.
i) For every positive integer n and a + ve rational number a
ii) If n is a positive integer and a, b are rational numbers then
iii) If n is a + ve integer and a, b are rational numbers then
iv) If m and n are positive integers and a is a positive rational number then
v) If m and n are positive integers and a is a positive rational number then
ORDER AND BASE OF SURDS
Surd is a surd of nth order with base a. is a quadratic surd
is a cubic surd
is a biquadratic surd
Pure surds and mixed surds. A surd which does not have a rational factor other than unity (1) is called a pure surd. For example etc. are pure surds.
A surd which has a rational factor other than unity, the other factor being rational is called a mixed surd e.g. etc are mixed surds.
Conversion of mixed surds into pure surds.
Example: Express each of the following as pure surds
To express the given surds as mixed surds in its simplest form
Example 1: Express each of the following as mixed surds in its simplest form
conversion of a surd into a surd of given order
step I Obtain the order of given surd say n
step II Obtain the order of new surd into which the given surd is to be converted say m
step III compute m/n let m/n = k
step IV write the new surd as
Convert into a surd of order 6
Solution: n = order of given surd = 2
m= order of new surd = 6
Let K = m/n = 6/2 = 3
Convert into a surd of order 8
Here n = 2
M = 8
K = m/n = 8/2 = 4.
Example 3: Convert into a surd of order 6
Here n = 3
M = 6
K = m/n = 6/3 = 2
Example 4: Express 2 x as pure Surd of order 6
Now n = 3
m = 6
So given surd =
Example 5: Express as pure surd of order 4
Here n = 2
M = 4
Conversion of two or more surds into surds of the same order.
Step 1.Let the surds be
So that then order are n1, n2, n3 …….. etc
Step II Find out LCM of n1, n2, n3 ………. xn
= n (say)
Step III Compute and
Step IV Write the requested surd as
Example 1: Convert into surds of the same but smallest order
Solution: n1 = order of =4
n2 = order of = 6
n = LCM of 4 and 6 – 12
Now m1 =
So required surds are
Example 2: Convert and into surds of same but smallest order
n1 = order of the surd
n2 = order of the surd
n = LCM of 2, 3, = 6.
Now m1 =
Thus reqd. surds are
Comparison of surds.
If the surds are of the same order, they can be compared by comparing their radicands.
Example 3: Which surd is larger or
Solution: Given surds are of the same order and their radicands are 26 and 35 respectively
Since 35 > 26 so
Example 4: Arrange the following surds in ascending order
Solution: Surds are of the same order
Comparing the surds of distinct order
We first reduce them to the same but smallest order and then compare their radiants.
Example 5 : Which surd is larger
Solution: Order of the two surds is 3, 4
Their LCM is 12
Example 6: Which is greater
n1 = 2 n2 = 3
n = LCM of 2, 3 = 6
Example 7: Arrange the following surds in ascending order of magnitude
Solution: Given surd are
The order of these surds is 2, 3, 6
LCM of 2, 3, 6 is 6
The surds can be written as
as it is
Now compare the radiants
ADDITION AND SUBTRACTION OF SURDS
Surds having same irrational factor are called similar surds. e.g. are similar surds. such type of surds can be added or subtracted. Where as unlike surds having no common irrational factor cannot be added or subtracted.
If the surds are unlike, reduce each of them to its simplest form and express each in such a way that they have a common irrational factor.
Example 1: Simplify
Example 2: Simplify
Example 3: Simplify 2
Multiplication of Surds
Surds of the same order can e multiplied
If the surds are not of the same order, they can be multiplied after converting them to surds of the same order
Division of Surds
a. Surds of the same order may be divided
i) If the product of two surds is a rational number then each one of them is called the rationalizing factor of the other.
are conjugate surds