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Formula 1. (ab)^2 a^2  2 abb^2

Formula 2.(ab)^2 a^2  2 abb^2

Remark : It is seen from formula 1 and formula 2 that, adding 2 ab or  2 abwith

a^2 b^2 , we get a perfect square, i.e. we get, (ab)^2 or (ab)^2. Substituting
binstead of b in formula 1, we get formula 2 :
{a(b)}^2 a^2  2 a(b)(b)^2
That is, (ab)^2 a^2  2 abb^2.

Corollary 1.a^2 b^2 (ab)^2  2 ab

Corollary 2. a^2 b^2 (ab)^2  2 ab
Corollary 3. (ab)^2 (ab)^2  4 ab

Proof : (ab)^2 a^2  2 abb^2
= a^2  2 abb^2  4 ab
= (ab)^2  4 ab
Corollary 4. (ab)^2 (ab)^2  4 ab
Proof : (ab)^2 a^2  2 abb^2
= a^2  2 abb^2  4 ab
= (ab)^2  4 ab

Corollary 5.
2

2 2 (a b)^2 (a b)^2

a b

  



Proof : From formula 1 and formula 2,

a^2  2 abb^2 (ab)^2

a^2  2 abb^2 (ab)^2

Adding, 2 a^2  2 b^2 (ab)^2 (ab)^2

or, 2 (a^2 b^2 ) (ab)^2 (ab)^2

Hence,

2

( ) ( )

( )

2 2

a^2 b^2 ab ab

Corollary 6.

2 2

2 2

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a b a b

ab

Proof : From formula 1 and formula 2,

a^2  2 abb^2 (ab)^2

a^2  2 abb^2 (ab)^2

Subtracting, 4 ab (ab)^2 (ab)^2