2.18 Algebraic Surds:
We have studied the surds and their representation
on number line in section 1.7.
In this section, the variables we are going to
use(x, a, b, n.) are all natural numbers.
In the surd represented by _{} we call m as the ‘order’ and n as the ‘radicand’
Definition: ‘Like surds’ are Group of surds which have same
order and same radicand after simplification (in their simplest form)
A group of surds of different order or different
radicand in their simplest
form are called ‘unlike surds’
Example: Let us observe
the following surds
1._{}
2._{}
If we had not simplified the above two surds, we
would have classified them as unlike surds. This is because their radicands
(48, 12) are not same, though the order is same. We need to compare surds only
after converting them to their simplest form.
Thus the above 2 surds are like surds as
their order=2 and radicand=3 in their simplest form
3._{}à order = 3, radicand=2
4._{} à order = 4, radicand=5
The above surds do not have same radicand and order
and hence they are called unlike surds.
Observe the way we do the following operations
1. 5a+3a =(5+3)a =8a
2. 7a2a =(72)a= 5a
We do additions/subtractions of surds in a similar
manner.
1. Sum or difference of like surds in the simplest form is
obtained by adding or subtracting their coefficients
2.18 Problem 1: Simplify _{}
Solution:
_{}
2.18 Problem 2. Simplify _{}
Solution:
_{}
=_{}=_{}= 2x^{ (1+1/2)}
= 2x^{3/2}
2.18 Problem 3. Subtract _{} from _{}^{}
Solution:
Result = (_{}) –(_{})
= _{}
=_{}
Observe following:
_{}
2. We followed the rule _{} similar to the rule (ab)^{ n}= a^{n} *b^{n}^{}
^{ }
2.18 Problem 4. Multiply _{} by _{}
Solution:
We know that _{}=5^{1/2}= 5^{2/4}= (5^{2})^{1/4}=
(25)^{1/4}
_{}_{} =(25)^{1/4}*
3^{1/4}= 75^{1/4}=_{}
What did we do?
Steps followed for multiplication of surds:
Step 1 : Write the surds in
index form.
Step 2 : Find LCM of
orders of given surds
Step 3 : Convert surds to
have equal orders
Step 4 : Multiply
radicands by following the rule_{}
2.18 Problem 5. Multiply _{} by _{}
Solution:
No 
Step 
Explanation 
1 
_{} = 3^{1/3}, _{}=2^{1/4} 
Write
the surds in index form. 
2 
The
orders of the surds are 3 and 4. Their LCM
is 12 
Find
LCM of orders 
3 
_{} = 3^{1/3}= 3^{4/12}
= (3^{4})^{1/12 }= (81)^{1/12 } 
Change
indices of surds 
4 
_{}=2^{1/4} = 2^{3/12 }=2^{3/12} = (2^{3})^{1/12
}= (8)^{1/12} 
Change
indices of surds 
5 
(_{}) *( _{}) = _{} 

We know that _{} is an irrational
number. How do we convert _{}to a rational number?
Let us multiply _{} by _{} then we have
_{} *_{} = _{} = 5. Note 5 is a rational number
Definition: The procedure of multiplying a surd by
another surd to get a rational number is called ‘Rationalisation’
The operands are called rationalizing
factor (RF) of the other.
In the above example _{} is RF of _{}
2.18 Problem 6 What is the RF of _{} ?
Solution:
Note that in the surd only _{} is irrational .It’s
co efficient 6 is rational number. Therefore we need to find RF only of _{} .
The RF of _{} is _{} because
_{}*_{} = _{} = (ab)
Now Multiply _{} by _{}
Result= 6(ab)^{1}^{/3}*((ab)^{2})^{1/3}
= 6(ab)^{1}^{/3}*(ab)^{2/3}
= 6(ab)^{(}^{1+2)/3
}=6(ab)
which is a rational number
Definition: A binomial surd is an algebraic sum (sum or
difference) of 2 terms both of which could be surds or one could be a rational
number and another a surd
Examples of Binomial surds are_{}, _{},_{}
RF or ‘Conjugate’
of a binomial surd is the term which when multiplied by the binomial surd,
results in a rational number.
(Conjugate of binomial surd* Binomial surd =
Rational number)
2.18 Problem 7 : Find the conjugate of _{}
Solution:
Note _{}= 2(_{})
We need to find a term such that the result has x
and y with rational coefficients
We also know (a+b)(ab) = a^{2}b^{2 } and hence _{} appears to be the conjugate of _{}
Therefore
_{} * _{}
= 2(_{})*(_{})
= 2{(_{})^{2}(_{})^{2}}^{}
= 2{2^{2}*(_{})^{2}(_{})^{2`}}
= 2(4xy) =8x2y
which is a rational number
For rationalization of
the surd in the denominator, we follow the following steps:
1) Find the RF of denominator
2) Multiply both numerator and denominator of surd
by RF of denominator
2.18 Problem 8: Rationalise denominator and simplify 2/(_{})
Solution:
Since (_{})*(_{}) =(xy) (_{}(ab)(a+b)
= a^{2}b^{2} with a =_{} b =_{})
We note that
_{} is conjugate of the
denominator. We multiply numerator and
denominator by this conjugate
_{}2/(_{})
={2/(_{})}*{(_{})/(_{})}
=2 (_{})/(xy)
2.18 Problem 9: Rationalise denominator and simplify (_{})/(_{})
Solution:
As in the above example
_{}is conjugate of _{}
_{}(_{})/(_{})
= {(_{})/(_{})}*{(_{})/(_{})}
= (_{})*(_{})/(9*25) (_{}(_{})^{2}=9*2 and (_{})^{2}=5)
= (_{})/13
=(_{})/13
= (_{})/13
2.18 Problem 10 : Rationalise
denominator and simplify 7_{}/ (_{})  _{}/ (_{})
Solution:
Let us rationalise the
terms separately
1. Multiply both numerator and denominator of the
first term by _{} which is conjugate of _{}
Note (_{})*(_{})= 103 =7
_{}7_{}/ (_{})
= 7_{}*(_{})/((_{})*(_{}))
= 7(_{})/7 (_{}_{}*_{} = 3)
= 3+_{}
2. Multiply both
numerator and denominator of the second term by_{} which is conjugate of _{}
_{}_{}/ (_{})
=_{}(_{})/(62)
=_{}/4
= _{}/2
_{}7_{}/ (_{})  _{}/ (_{})
= (3+_{})  _{}/2
= (6+ 2_{} _{}+_{})/2
= (6+_{}+_{})/2
2.18 Summary of
learning
No 
Points to remember 
1 
_{} 
2 
Rationsalisation
is a process of finding a term such
that the product of this term and the surd is
a rational number 