Rationalization of Denominator

We at ask-math believe that educational material should be free for everyone. Please use the content of this website for in-depth understanding of the concepts. Additionally, we have created and posted videos on our youtube.

We also offer One to One / Group Tutoring sessions / Homework help for Mathematics from Grade 4th to 12th for algebra, geometry, trigonometry, pre-calculus, and calculus for US, UK, Europe, South east Asia and UAE students.

Affiliations with Schools & Educational institutions are also welcome.

Please reach out to us on [email protected] / Whatsapp +919998367796 / Skype id: anitagovilkar.abhijit

We will be happy to post videos as per your requirements also. Do write to us.

In this section we will discuss rationalization of denominator.

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) = a² - b²]
= (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

From real- number to number system

Home Page

Russia-Ukraine crisis update - 3rd Mar 2022

The UN General assembly voted at an emergency session to demand an immediate halt to Moscow's attack on Ukraine and withdrawal of Russian troops.