# Rationalization of Denominator

**Covid-19 has led the world to go through a phenomenal transition .**

**E-learning is the future today.**

**Stay Home , Stay Safe and keep learning!!!**

Sometimes we come across expressions containing square roots in their denominators. In such an expression if the denominator is free from square roots then it will be easier to add/subtract/multiply or divide. To make the denominators free from square roots, we multiply the numerator and denominator by an irrational number. Such a number is called

**rationalization factor.**

**Example :**(1 - √3 ) / √2

As there is √2 in the denominator and we know that √2 x √2 = 2

So, multiply top and bottom by √2 .

[(1-√3) x √2] / (√2 x √2)

= [1√2 - √2 x √3] / 2 [ use distributive property]

=[ √2 - √6] /2

Note : Rationalization factor for :
1) 1/√a ------> √a 2) a + √b -------> a - √b 3) a - √b ---------> a + √b 4) √a + √b ----------> √a - √b 5) √a - √b ------------> √a + √b |

**Examples on rationalization of denominator**

1) Rationalise the denominator of 2/√3

**Solution :**

We know that the rationalization factor for 1/√a is √a .

∴ 2/√3 = (2 x √3)/ (√3 x √3)

= 2√3/3

-----------------------------------------------------------------

2) Rationalise the denominator of 1/(3 - √2)

**Solution :**

We have,

1/(3 - √2) = 1(3 + √2) /(3 - √2)(3 + √2)

= (3 + √2)/( 9 – 2) [ use the identity of (a+b)(a-b) = a

^{2}- b

^{2}]

= (3 + √2 )/ 7

-----------------------------------------------------------------

3) Solve : 3/(√3 + 1) + 5/(√3 – 1)

**Solution :**

3/(√3 + 1) + 5/(√3 – 1)

Rationalize the each term and then solve.

3/(√3 + 1) = 3 (√3 -1)/( √3 +1)( √3 -1)

=(3√3 – 3)/(3-1) = (3√3 – 3)/2 --------> (1)

5/(√3 – 1) = 5(√3 + 1)/( √3 -1)( √3 +1)

= (5√3 + 5)/(3-1) = (5√3 + 5)/2 --------> (2)

Add equation (1) and (2) we get,

(3√3 – 3)/ 2 + (5√3 + 5 )/2

= ( 3√3 – 3 + 5√3 + 5)/2

= (8√3 + 2)/2

= 2(4√3 + 1)/2

= 4√3 + 1

-----------------------------------------------------------------

4) √7(√35 - √7) = a + b√5 , find the value of a and b.

**Solution :**

√7(√35 - √7) = a + b√5

√7(√(7 x5) - √7) = a + b√5

√7(√7 x √5 - √7) = a + b√5

√7 x √7 x √5 - √7 x√7 = a + b√5 [ use a distributive law]

7√5 – 7 = a + b√5

-7 + 7√5 = a +b√5

∴ a = - 7 and b = 7.

**Real-Numbers**

• Real Numbers

• Representation of real-numbers on number line

• Operations on Real Numbers

• Rationalization of denominator

• Real Numbers

• Representation of real-numbers on number line

• Operations on Real Numbers

• Rationalization of denominator

Home Page

**Covid-19 has affected physical interactions between people.**

**Don't let it affect your learning.**