The rationalized value of √5/(√6 + √2) is (√30 – √10)/4. The detailed solution for the same is added below:

Rationalize Denominator of √5/(√6 + √2)

Solution:

Given, √5/(√6 + √2)

To rationalize the denominator, multiply and divide the given term with (√6 – √2).

= √5/(√6 + √2) × (√6 – √2)/(√6 – √2)

Since, (a + b)(a – b) = a² – b²

(√6 + √2)×(√6 – √2) = (√6)² – (√2)² 

= 6 – 2 = 4

 √5/(√6 + √2) × (√6 – √2)/(√6 – √2) = √5 × (√6 – √2)/4

= [√(5×6) – √(5×2)]/4

= (√30 – √10)/4

Hence, √5/(√6 + √2) = (√30 – √10)/4

Practise Problems

Problem 1: Rationalize: 1/(√2 − √7). 

Solution:

Given: 1/(√2 − √7).

To rationalize the denominator, multiply and divide the given term with (√2 + √7).

= 1/(√2 − √7) × (√2 + √7)/(√2 + √7)

Since, (a + b)(a – b) = a² – b²

(√2 − √7)×(√2 + √7) = (√2)² − (√7)² 

= 2 − 7 = −5

1/(√2 – √7) × (√2 + √7)/(√2 + √7)

= (√2 + √7)/(-5)

= −(√2 + √7)/5

Hence, 1/(√2 – √7) = −(√2 + √7)/5

Problem 2: Rationalize: 1/(4 − √6).

Solution:

Given: 1/(4 − √6).

To rationalize the denominator, multiply and divide the given term with (4 + √6).

= 1/(4 − √6) × (4 + √6)/(4 + √6)

Since, (a + b)(a – b) = a² – b²

(4 − √6)×(4 + √6) = (4)2 − (√6)2

= 16 −6 = 10

1/(4 − √6) × (4 + √6)/(4 + √6) = (4 + √6)/10

Hence, 1/(4 − √6) = (4 + √6)/10.

Problem 3: Rationalize: a) 3/√5   b) 5 / ³√4.

Solution:

• 3/√5

To rationalize the denominator, multiply and divide the given term with √5.

= 3/√5 × √5/√5

= 3√5/5

Hence, 3/√5 = 3√5/5.

• 5 / ³√4 = 5/(4)^1/3

To rationalize the denominator, multiply and divide the given term with 4^2/3.

= 5/(4)^1/3 × 4^2/3/4^2/3

= 5 × 4^2/3  /4^{(1/3 + 2/3)}

= 5 × (4²)^1/3/4

= 5/4 × 4^2/3

Thus, 5/³√4 =  5/4 × 4^2/3

Problem 4: Rationalize: −3/(1 + √2).

Solution:

Given: −3/(1 + √2)

To rationalize the denominator, multiply and divide the given term with (1 − √2).

= −3/(1 + √2) × (1 − √2)/(1 − √2)

Since, (a + b)(a – b) = a² – b²

(1 + √2)×(1 − √2) = (1)2 − (√2)2

= 1 − 2 = − 1

−3/(1 + √2) × (1 − √2)/(1 − √2) = −3(1 − √2)/(−1)

= 3(1 − √2)

Hence, −3/(1 + √2) = 3(1 − √2).

Problem 5: Rationalise: −6/√3

Solution:

 −6/√3

To rationalize the denominator, multiply and divide the given term with √3.

= −6/√3 × √3/√3

= −6√3/3

Hence,

−6/√3 = −6√3/3.

Frequently Asked Questions

How to rationalize the denominator that contains a square root?

 To rationalise the denominator that contains a square root, we multiply and divide the given rational number with the same square root value. In this way, the denominator changes to a rational number.

What is meant by rationalization of denominator?

Rationalisation of denominator signifies to remove any radical term or surds from the denominator and expressing the rational number in a simplified form.

What value cannot be used in the denominator?

Denominator of any rational number cannot be zero, as zero in denominator is undefined number.

How do we rationalize the denominator which contains two terms?

For rationalising the denominator that contains two terms, firstly we find the conjugate pair of the denominator and then multiply and divide the fraction by the sme conjugate pair of the denominator.