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

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. 7a-2a =(7-2)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 co-efficients

2.18 Problem 1: Simplify

Solution:

2.18 Problem 2. Simplify

Solution:

=== 2x (1+1/2)

= 2x3/2

2.18 Problem 3. Subtract  from

Solution:

Result = () –()

=

=

Observe following:

2. We followed the rule  similar to   the rule (ab) n= an *bn

2.18 Problem 4. Multiply  by

Solution:

We know that =51/2= 52/4= (52)1/4= (25)1/4

=(25)1/4* 31/4= 751/4=

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 = 31/3, =21/4 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 = 31/3= 34/12 = (34)1/12 = (81)1/12 Change indices of  surds 4 =21/4 = 23/12 =23/12 = (23)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

* =  = (a-b)

Now Multiply  by

Result= 6(a-b)1/3*((a-b)2)1/3

= 6(a-b)1/3*(a-b)2/3

= 6(a-b)(1+2)/3 =6(a-b)  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 co-efficients

We also know (a+b)(a-b) = a2-b2  and hence  appears to be the  conjugate of

Therefore

*

= 2()*()

= 2{()2-()2}

= 2{22*()2-()2`}

= 2(4x-y) =8x-2y 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 ()*() =(x-y) ((a-b)(a+b) = a2-b2  with  a  = b =)

We note that   is conjugate of the denominator.  We multiply numerator and denominator by this conjugate

2/()

={2/()}*{()/()}

=2 ()/(x-y)

2.18 Problem 9: Rationalise denominator and simplify ()/()

Solution:

As in the above example is conjugate of

()/()

= {()/()}*{()/()}

= ()*()/(9*2-5)                (()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 ()*()= 10-3 =7

7/ ()

= 7*()/(()*())

= 7()/7          (* = 3)

= 3+

2. Multiply both numerator and denominator of the second term by which is conjugate of

/ ()

=()/(6-2)

=/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