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Question
Evaluate `(sqrt3 + sqrt2)^6 - (sqrt3 - sqrt2)^6`
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Solution
`(a + b)^6 = ^6C_0 a^6 + ^6C_1 a^5 b + ^6C_2 a^4 b^2 + ^6C_3 a^3 b^3 + ^6C_4 a^2 b^4 + ^6C_5a^1b^5 + ^6C_6 b^6`
= `a^6 + 6a^5b + 15a^4 b^2 + 20a^3 b^3 + 15a^2 b^4 + 6ab^5 + b^6`
`(a - b)^6 = ^6C_0 a^6 - ^6C_1 a^5 b + ^6C_2 a^4 b^2 - ^6C_3 a^3 b^3 + ^6C_4 a^2 b^4 - ^6C_5a^1b^5 + ^6C_6 b^6`
= `a^6 - 6a^5b + 15a^4 b^2 - 20a^3 b^3 + 15a^2 b^4 - 6ab^5 + b^6`
∴ `(a + b)^6 - (a -b)^6 = 2(6a^5b + 20a^3 b^3 + 6ab^5)`
Putting a = `sqrt3` and b = `sqrt2`, we obtain
`(sqrt3 + sqrt2)^6 - (sqrt3 + sqrt2)^6` = `2[6(sqrt3)^5 (sqrt2) + 20 (sqrt3)^3 (sqrt2)^3 + 6 (sqrt3)(sqrt2)^5]`
= `2[54sqrt6 + 120 sqrt6 + 24 sqrt6]`
= `2 xx 198 sqrt6`
= `396 sqrt6`
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