# Rationalization of Denominators

Rationalization of Denomintors is a method where we change the fraction with an irrational denominator into a fraction with a rational denominator. If there is an irrational or radical in the denominator the definition of rational number ceases to exist as we can’t divide anything into irrational parts. Thus, the rationalization process comes in handy to convert these fractions with irrational denominators into fractions of integer denominators.

It is also important to understand that rationalization does not change the value of a number or function. It is a technique to re-write the fraction in a more acceptable form that is easy to understand. Readers may use a calculator to confirm that rationalization does not change the original value in the further examples used for the explanation of the method.

## Rationalization Definition

As the name suggests, rationalization is a process to make a fraction rational. Rationalization is a process by which radicals in the denominator of a fraction are removed by multiplying it with an irrational number generally a conjugate or a similar radical. Rationalization makes the denominator free from radicals like square root or cube root.

## Rationalizing Factor

The number or expression, by which the denominator is multiplied to convert it into rational is called the Rationalizing Factor. Some of the Rationalizing factors are tabulated below:

## How to Rationalize the Denominator?

As different forms of irrational denominators need different methods to rationalize. Thus all the various methods to Rationalize the Denominator are as follows:

### Rationalizing Single-Term Denominator

To rationalize a monomial square or cube root say  where n < m, we multiply the numerator and denominator by the same factor say     and we get    which can be replaced by y, so free from the radical term. Or in other words, to rationalize a monomial square or cube root, we multiply the numerator and denominator by the same factor as the denominator. i.e., if we have  as denominator, then we need to multiply with

Example: Let us rationalize 1/âˆš5

Solution:

1/âˆš5 is the given expression,

So, multiple both numerator and denominator byâˆš5

= 1/âˆš5 Ã— âˆš5/âˆš5

= âˆš5/5

Example: Rationalize .

Solution:

is the given expression,

So, multiple both numerator and denominator by

### Rationalizing Two Terms Denominator

If the denominator is linear and is of the form a +âˆšb or a + iâˆšb, then the method of rationalization of the denominator comprises multiplying both the numerator and the denominator by the algebraic conjugate a – âˆšb or a – iâˆšb. Due to the result of the algebraic identity (a+b)(a-b) = a2 – b2, the denominator of form a +âˆšb or a + iâˆšb can always be rationalized using this method.

Example: Let us rationalize 1/(1 +âˆš5)

Solution:

Given expression is 1/(1 +âˆš5).

So, multiple both numerator and denominator by 1 – âˆš5

[Using identity (a+b)(a-b) = a2 – b2]

### Rationalizing Three Terms Denominator

If the denominator is trinomial, like aÂ±âˆšbÂ±âˆšc (Â± represent all the possibilities) then it’s a little more complicated to rationalize its denominator than the method of rationalization of fraction with binomial radical as its denominator. In this case, take two terms as a single term and the third term as a second term of the rationalizing factor and then rationalize. If irrational terms are not eliminated completely after the first process of rationalization then rationalize the obtained result again with the term that remained irrational in the first rationalization process.

Let’s take 1/(1+âˆš2-âˆš3) for example,

Step 1: Choose any of the two given radicals to make it look like a binomial radical and multiply the fraction with its conjugate,

1/(1+âˆš2-âˆš3) Ã— (1+âˆš2+âˆš3)/(1+âˆš2+âˆš3)

Note: if you choose (1+âˆš2)-âˆš3, then multiply the denominator and numerator with (1+âˆš2)+âˆš3 and if you choose (1-âˆš3)+âˆš2, then multiply the denominator and numerator with (1-âˆš3)-âˆš2.

Step 2: Simplify.

(1+âˆš2+âˆš3)/[(1+âˆš2)2-(âˆš3)2]

(1+âˆš2+âˆš3)/(3+2âˆš2-3)

(1+âˆš2+âˆš3)/2âˆš2

Step 3: Multiply again with âˆš2 in numerator and denominator,

(1+âˆš2+âˆš3)/2âˆš2 Ã— âˆš2/âˆš2

Step 4:  Simplify the result in Step 3.

(âˆš2+2+âˆš6)/4

## Sample Problems on Rationalization of Denominators

Problem 1: What is the interpretation of  1/âˆš3 on a number line?

Solution:

Since the denominator has square root in the denominator, it is a bit difficult to understand. As we can not divide anything is âˆš3 parts as it is an irrational number and it exact location on the number line depends on the number of digits taken into consideration at a time.

Let us write an equivalent expression where the denominator is a rational number using the method of rationalization.

Multiply and divide the given expression by âˆš3.

= 1/âˆš3 Ã— âˆš3/âˆš3

= âˆš3/3

Thus, 1/âˆš3 =âˆš3/3 means a point which is at one third distance from 0 to âˆš3.

So, we can interpret the meaning of 1/âˆš3 as a point which lies at one third distance from 0 to âˆš3.

Problem 2:  Rationalize the denominator (3 +âˆš7)/âˆš7

Solution:

As given expression is (3 +âˆš7)/âˆš7.

Multiply and divide the given expression by âˆš7.

= (3 + âˆš7)/âˆš7 Ã— (âˆš7/âˆš7)

= ((3 + âˆš7)Ã—âˆš7 )/âˆš7Ã—âˆš7

= (3âˆš7 + 7)/7

Problem 3: Find the value of a and b, If 1/(5 + 6âˆš3) = aâˆš3 + b.

Solution:

Given: 1/(5 + 6âˆš3) = aâˆš3 + b.

Taking LHS = 1/(5 + 6âˆš3)

Multiply and divide the given expression by 5 – 6âˆš3 to rationalize it.

â‡’ LHS ={1/(5 + 6âˆš3)} * {(5 – 6âˆš3)/(5 – 6âˆš3}

â‡’ LHS= {1 Ã—(5 – 6âˆš3)}/{(5 + 6âˆš3)(5 – 6âˆš3)}

Using the identity (a + b)(a – b) = a2 – b2

â‡’ LHS = (5 – 6âˆš3)/{52 – (6âˆš3)2}

â‡’ LHS =(5 – 6âˆš3)/ 25 – 108

â‡’ LHS= (5 – 6âˆš3)/ -83

â‡’ LHS = (6âˆš3 – 5)/83

Given that 1/(5 + 6âˆš3) = aâˆš3 + b

â‡’ (6âˆš3 – 5)/83 = aâˆš3 + b

â‡’ a = 6/83, b = -5/83

Problem 4: Given that âˆš5 = 2.236. Find the value of 3/âˆš5

Solution:

As given expression is 3/âˆš5 .

Multiply and divide the given expression by âˆš5

=(3/âˆš5) Ã— (âˆš5 /âˆš5)

= 3 âˆš5 /5

= (3/5) Ã—âˆš5

= 0.6 Ã— 2.236    [ Given âˆš5 = 2.236]

= 1.3416

Thus, 3/âˆš5 = 1.3416

Problem 5: Rationalize the denominator of 8/(âˆš5 – âˆš3)

Solution:

As given Expression is 8/(âˆš5 – âˆš3)

Multiply and divide the given expression by âˆš5 + âˆš3

= (8 Ã—(âˆš5 + âˆš3))/((âˆš5 – âˆš3)âˆš5 + âˆš3))

Using the identity (a + b)(a – b) = a2 – b2

= (8âˆš5 + 8âˆš3)/(âˆš52 – âˆš32)

= 8âˆš5 + 8âˆš3/(5 – 3)

= 8âˆš5 + 8âˆš3/2

= 4âˆš5 + 4âˆš3

Problem 6: Simplify: (2âˆš2 + âˆš6 – âˆš3)/(âˆš2 – âˆš3 + âˆš6)

Solution:

As given expression if (2âˆš2 + âˆš6 – âˆš3)/(âˆš2 – âˆš3 + âˆš6).

To rationalize this expression, multiply the numerator and denominator by (âˆš2 + âˆš6 + âˆš3).

= [(2âˆš2 + âˆš6 – âˆš3)/(âˆš2 – âˆš6 + âˆš3)] Ã— [(âˆš2 + âˆš6+ âˆš3)/(âˆš2 + âˆš6 + âˆš3)]

= [2âˆš2(âˆš2 + âˆš6 + âˆš3) + âˆš6(âˆš2 + âˆš6 + âˆš3) – âˆš3(âˆš2 + âˆš6 + âˆš3)]/[âˆš2(âˆš2 + âˆš6 + âˆš3) – âˆš3(âˆš2 + âˆš6 + âˆš3) + âˆš6(âˆš2 + âˆš6 + âˆš3)]

Simplifying the numerator and denominator:

[(4âˆš2 + 2âˆš6 – âˆš3âˆš2 – âˆš3âˆš3 – âˆš3âˆš6)/(2 – 3 + 6)] Ã— (âˆš2 + âˆš3 + âˆš6)

Simplifying the first part of the expression:

[4âˆš2 + 2âˆš6 – âˆš6 – âˆš9 – âˆš18]/5

[4âˆš2 + 2âˆš6 – âˆš6 – 3âˆš2 – 3âˆš2]/5

[âˆš2 – âˆš6]/5

So, the simplified expression is (âˆš2 – âˆš6)/5.

## FAQs on Rationalization of Denominators

### Q1: What is the Rationalization of Denominators?

Rationalization of denominator is the process to convert denominator of the fraction with irrational or radical expression

### Q2: Why is it important to Rationalize Denominators?

Fractions can represent parts of a whole, but when there is an irrational or radical in the denominator, the definition of the fraction no longer applies. To deal with irrational or radical denominators rationalization is used to cover these fractions into fractions with integer denominators.

### Q3: What is Conjugate?

Conjugate is the binomial formed by changing the sign of the given binomial. For example, a-âˆšb is the conjugate of a+âˆšb.

### Q4: How do you Rationalize a Denominator With 2 Terms?

To rationalize a denominator with 2 terms, we need to multiply both the numerator and denominator by the conjugate of the denominator and simplify.

### Q5: How to Rationalize a Denominator with Square Root?

We have to multiply the numerator and denominator with the same square root term to rationalize the denominator with square root.