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Question
Find (a + b)4 – (a – b)4. Hence, evaluate `(sqrt3 + sqrt2)^4 - (sqrt3 - sqrt2)^4`
Solution
Using Binomial Theorem, the expressions, (a + b)4 and (a – b)4, can be expanded as
`(a + b)^4 = ^4C_0 a^4 + ^4C_1 a^3 b + ^4C_2 a^2b^2 + ^4C_3 ab^3 + ^4C_4 b^4`
(a - b)4 = 4C0 a4 - 4C1 a3b + 4C2 a2b2 - 4C3 ab3 + 4C4b4
∴ `(a + b)^4 - (a - b)^4 = ^4C_0 a^4 + ^4C_1 a^3 b + ^4C_2 a^2b^2 + ^4C_3 ab^3 + ^4C_4 b^4`
[4C0 a4 - 4C1 a3b + 4C2 a2b2 - 4C3 ab3 + 4C4 b4]
2 (4C1a3b + 4C3ab3) = 2(4a3b + 4ab3)
= 8ab (a2 + b2)
In this, by substituting `a = sqrt 3 , b = sqrt 2`
`(sqrt3 + sqrt2)^4 - (sqrt3 - sqrt2)^4`
= `8sqrt3. sqrt2 [(sqrt3)^2 + (sqrt2)^2]`
= `8sqrt6 (3 + 2) = 40sqrt6`
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