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  • Find (a + b)^4 – (a – b)^4 . Hence, evaluate (root of 3 + root of 2 )^4 – (root of 3 – root of 2 )^4 .

Q11. Find (a + b)^4 - (a-b)^4. Hence, evaluate(\sqrt{3} + \sqrt2)^4 - (\sqrt3-\sqrt2)^4.

Answers (1)

Using the binomial theorem, the expressions (a+b)^4 and (a-b)^4 can be expressed as

(a+b)^4=^4C_0a^4+^4C_1a^3b+^4C_2a^2b^2+^4C_3ab^3+^4C_4b^4

(a-b)^4=^4C_0a^4-^4C_1a^3b+^4C_2a^2b^2-^4C_3ab^3+^4C_4b^4

From here,

(a+b)^4-(a-b)^4 = ^4C_0a^4+^4C_1a^3b+^4C_2a^2b^2+^4C_3ab^3+^4C_4b^4-^4C_0a^4+^4C_1a^3b-^4C_2a^2b^2+^4C_3ab^3-^4C_4b^4

(a+b)^4-(a-b)^4 = 2\times( ^4C_1a^3b+^4C_3ab^3)

(a+b)^4-(a-b)^4 = 8ab(a^2+b^2)

Now, using this, we get 

(\sqrt{3} + \sqrt2)^4 - (\sqrt3-\sqrt2)^4=8(\sqrt{3})(\sqrt{2})(3+2)=8\times\sqrt{6}\times5=40\sqrt{6}

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neha

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