Simplifying radical expressions using conjugates :

Whenever we want to simplify radical expressions using conjugates, first we have to consider the denominator. If the denominators are in one of following forms, then we have to do the following steps for simplification.

How to simplify radical expressions using conjugates?

Case 1 :

If the denominator is in the form of a ± √b or a  ± c √b  (where b is a rational number).

Then we have to multiply both the numerator and denominator by its conjugate.

a + √b and a - √b are conjugate of each other.

a + c√b and a - c√b are conjugate of each other.

Case 2 :

If the denominator is in the form of ± √b (where a and b are rational numbers).

Then we have to multiply both the numerator and denominator by its conjugate.

a + √b and a - √b are conjugate of each other.

What is a radical conjugate ?

If the product of two irrational numbers is rational, then each one is called the radical conjugate  of each other.

Simplifying radical expressions using conjugates - Examples

Example  1 :

Rationalize the denominator 1/(2 + √5)

Solution :

Because we have the denominator 2 + √5, we have to multiply by its conjugate 2 - √5. By multiplying the numerators, we get 2 - √5 in denominator we get (2 + √5) (2 - √5). Comparing the denominator with the algebraic identity (a + b) (a - b), we have "2" instead of "a" and "√5" instead of b.

(a + b) (a - b)  =  a² - b²

That is,

(2 + √5) (2 - √5)  =  2² - √5²

=  4 - 5

=  -1 By distributing negative sign to the numerator, we get -2 + √5

Hence the answer is -2 + √5.

Example 2 :

Rationalize the denominator (6 + √5)/(6 - √5)

Solution :

Step 1 :

Because we have the denominator (6 - √5), we have to multiply by its conjugate (6 + √5)

=  [(6 + √5)/(6 - √5)] x [(6 + √5)/(6 + √5)]

Step 2 :

By multiplying the numerators, we get (6 + √5) (6 + √5), in denominator we get (6 - √5) (6 + √5)

=  [(6 + √5)²/(6 - √5)(6 + √5)]

Step 3 :

Comparing the denominator with the algebraic identity (a + b)², we have "6" instead of "a" and "√5" instead of b.

(a + b)²  =  a² + 2ab + b²

(6 + √5)²  =  6² + 2(6) (√5) + √5²

=  36 + 12√5 + 5

=  41 + 12√5

Comparing the denominator with the algebraic identity (a + b) (a - b), we have "6" instead of "a" and "√5" instead of b.

(a + b) (a - b)  =  a² - b²

That is,

(6 - √5)(6 + √5)  =  6² - √5²

=  36 - 5

=  31

Step 4 :

Hence the answer is (41 + 12√5) / 31

Example 3 :

Rationalize the denominator 1/(8 - 2 √5)

Solution :

Step 1 :

Because we have the denominator (8 - 2 √5), we have to multiply by its conjugate (8 + 2 √5)

=  [1/(8 - 2 √5)] x [(8 + 2 √5)/(8 + 2 √5)]

Step 2 :

By multiplying the numerators, we get (8 + 2 √5), in denominator we get (8 + 2 √5) (8 - 2 √5)

Step 3 :

Comparing the denominator with the algebraic identity (a + b) (a - b), we have "8" instead of "a" and "2√5" instead of b.

(a + b) (a - b)  =  a² - b²

That is,

(8 + 2 √5)  (8 - 2 √5)  =  8² - (2√5)²

=  64 - 2²√5²

=  64 - 4(5)

=  64 - 20

=  44

Step 4 :

=  (8 + 2 √5) / 44

By factoring 2, we get 2(4 - √5) / 44

=  (4 + √5) / 22

Hence the answer is (4 + √5) / 22

After having gone through the stuff given above, we hope that the students would have understood "Simplifying radical expressions using conjugates".

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