SIMPLIFYING RADICAL EXPRESSIONS USING CONJUGATES

Radical Conjugate

If the product of two irrational numbers is rational, then  the two irrational numbers are radical conjugate to the other.

Example : 

The product of two irrational numbers √2 and √8 is the rational number 4. 

That is, 

√2 ⋅ √8  =  √(2 ⋅ 8) 

√2 ⋅ √8  =  √16 

√2 ⋅ √8  =  √(4 ⋅ 4)

√2 ⋅ √8  =  4

So, √2 and √8 are radical conjugate to each other. 

Example 1 :

Simplify : 

1 / (2 + √5)

Solution :

Simplifying the above radical expression is nothing but rationalizing the denominator. 

So, rationalize the denominator. 

Here, the denominator is 2 + √5. 

In the given fraction, multiply both numerator and denominator by the conjugate of 2 + √5. That is 2 - √5.

Example 2 :

Simplify :

(6 + √5) / (6 - √5)

Solution :

Simplifying the above radical expression is nothing but rationalizing the denominator. 

So, rationalize the denominator. 

Here, the denominator is 6 - √5. 

In the given fraction, multiply both numerator and denominator by the conjugate of 6 - √5. That is 6 + √5.

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

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

(6 + √5) / (6 - √5)  =  (6 + √5)² / [6² - (√5)²]

(6 + √5) / (6 - √5)  =  [6² + 2(6)(√5) + (√5)²] / (36 - 5)

(6 + √5) / (6 - √5)  =  [36 + 12√5 + 5] / 31

(6 + √5) / (6 - √5)  =  (41 + 12√5) / 31

Example 3 :

Simplify : 

1 / (8 - 2√5)

Solution :

Simplifying the above radical expression is nothing but rationalizing the denominator. 

So, rationalize the denominator. 

Here, the denominator is 8 - 2√5. 

In the given fraction, multiply both numerator and denominator by the conjugate of 8 - 2√5. That is 8 + 2√5.

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

1 / (8 - 2√5)  =  (8 + 2√5) / [8² - (2√5)²]

1 / (8 - 2√5)  =  (8 + 2√5) / [64 - (4 ⋅ 5]

1 / (8 - 2√5)  =  (8 + 2√5) / [64 - 20]

1 / (8 - 2√5)  =  (8 + 2√5) / 44

1 / (8 - 2√5)  =  2(4 + √5) / 44

1 / (8 - 2√5)  =  (4 + √5) / 22

Example 4 :

Simplify : 

2 / √3

Solution :

Simplifying the above radical expression is nothing but rationalizing the denominator. 

So, rationalize the denominator. 

Here, the denominator is √3. 

In the given fraction, multiply both numerator and denominator by √3.

2 / √3  =  (2√3) / (√3 ⋅ √3)

2 / √3  =  2√3 / 3

Example 5 :

Simplify : 

1/√2  +  1/√5

Solution :

To add the above two fractions, make the denominators same. 

Least common multiple of √2 and √5 is 

=  √2 ⋅ √5

=  √(2 ⋅ 5)

=  √10

Then, 

1/√2  +  1/√5  =  √5/√10  +  √2/√10

1/√2  +  1/√5  =  (√5 + √2) / √10

To rationalize the denominator on the right side, multiply both numerator and denominator by √10.

1/√2  +  1/√5  =  [(√5+√2)√10] / (√10 ⋅ √10)

1/√2  +  1/√5  =  (√50 + √20) / 10

1/√2  +  1/√5  =  (√(5 ⋅ 5 ⋅ 2) + √2 ⋅2 ⋅ 5) / 10

1/√2  +  1/√5  =  (5√5 + 2√5) / 10

Example 6 :

 [√5/(√6 + 2)] - [√5/(√6 - 2)]

Solution :

=  [√5/(√6 + 2)]

= √5/(√6 - 2)/(√6 - 2)(√6 - 2)

= (√30 - 2√5)/(6 - 4)

= (√30 - 2√5)/2 -----(1)

=  [√5/(√6 - 2)]

= √5/(√6 - 2)/(√6 + 2)(√6 + 2)

= (√30 + 2√5)/(6 - 4)

= (√30 + 2√5)/2 -----(2)

(1) + (2)

= (√30 - 2√5)/2 + (√30 + 2√5)/2

= 2√30/2

= √30

Example 7 :

Find the value of a and b if

Solution :

= (√7 - 2)/(√7 + 2)

Multiplying both numerator and denominator by √7 - 2, we get

= [(√7 - 2)/(√7 + 2)] [(√7 - 2)/(√7 - 2)]

= (√7 - 2)(√7 - 2)/(√7 - 2)(√7 + 2)

= (7 - 4√7 + 4)/(7 - 4)

= (11 - 4√7)/3

So, the value of b is 11/3 and a is -4/3.

Example 8 :

If x = √5 + 2, then find the value of x² + 1/x²

Solution :

x = √5 + 2

x² = (√5 + 2)²

= 5 + 2√5 + 4

= 9 + 2√5 -----(1)

(1/x)² = [1/(√5 + 2)]²

= [(√5 - 2)/(√5 - 2)(√5 + 2)]²

= (√5 - 2)²/(5 - 4)

= 9 - 2√5-----(2)

(1) + (2)

= 18

Example 9 :

Given that √2  =  1.414, find the value of (8 - 5√2)/(3 - 2√2) (to 3 places of decimals).

Solution :

= (8 - 5√2)/(3 - 2√2)

Conjugate of (3 - 2√2) is (3 + 2√2)

= (8 - 5√2)(3 + 2√2)/(3 - 2√2)(3 + 2√2)

=  (24 + 16√2 - 15√2 - 20)/9-8

= 4 + √2

= 4 + 1.414

= 5.414

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