Rationalization is the process of eliminating radicals or imaginary numbers from the denominator of an algebraic expression. In the process of rationalisation, remove the radicals in a fraction so that the denominator only contains rational numbers.


Some Identities


  • (a + b)2 = a2 + 2ab + b2
  • (a – b)2 = a2 – 2ab + b2
  • a2 – b2 = (a + b)(a – b)
  • (x + a)(x + b) = x2 + (a + b) x + ab
  • (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
  • (a + b)3 = a3 + b3 + 3ab (a + b)
  • (a – b)3 = a3 – b3 – 3ab (a – b)
  • a3 + b3 + c3– 3abc = (a + b + c)(a2 + b2 + c2 – ab – bc – ca)



ILLUSTRATIVE EXAMPLES



Simplify the following expression:

(i) (4 + √7) (3 + √2)
= 12 + 4√2 + 3√7 + √14

(ii) (3 + √3)(5- √2 )
= 15 – 3√2 + 5√3 – √6

(iii) (√5 -2)( √3 – √5)
= √15 – √25 – 2√3 + 2√5
= √15 – 5 – 2√3 + 2√5

1. Using Identity: (a – b)(a+b) = a2 – b2
(i) (11 + √11) (11 – √11)
= 112 – (√11)2
= 121 – 11
= 110
(ii) (5 + √7) (5 –√7 )
= (52 – (√7)2 )
= 25 – 7 = 18
(iii) (√8 – √2 ) (√8 + √2 )
= (√8)2 – (√2 ) 2
= 8 -2
= 6
(iv) (3 + √3) (3 – √3)
= (3)2 – (√3)2
= 9 – 3
= 6
(v) (√5 – √2) (√5 + √2)
=(√5)2 – (√2)2
= 5 – 2
= 3
2. Using identities: (a – b)2 = a2 + b2 – 2ab and (a + b)2 = a2 + b2 + 2ab
(i) (√3 + √7)2
= (√3)2 + (√7)2 + 2(√3)( √7)
= 3 + 7 + 2√21
= 10 + 2√21

(ii) (√5 – √3)2
= (√5)2 + (√3)2 – 2(√5)( √3)
= 5 + 3 – 2√15
= 8 – 2√15
(iii) (2√5 + 3√2 )2
= (2√5)2 + (3√2 )2 + 2(2√5 )( 3√2)
= 20 + 18 + 12√10
= 38 + 12√10


Rationalisation - Class 9th - Notes






Rationalisation - Class 9th - RD Sharma Exercises with Solution