Question

Evaluate$( √3 +√2) ^6 - ( √3 -√2) ^6.$

Answer 1

Firstly, the expression (a+ b)6 - (a - b)6 is simplified by using Binomial Theorem.
This can be done as
$(a+b)^6 = C_0^6a^6 + C_1^61a^5b + C_2^6a^4b^2 + C_3^6a^3b^3 + C_4^6a^2b^4 + C_5^6a^1b^5 + C_6^6b^6$
$= a^6 + 6a^5b +15a^4b^2 + 20a^3b^3 + 15a^2b^4 + 6ab^5 + b^6$
$(a-b)^6 = C_0^6a^6 - C_1^6a^5b + C_2^6a^4b^2 - C_3^6a^3b^3 + C_4^6a^2b^4 - C_5^6a^1b^5 + C_6^6b^6$
$= a^6 - 6a^5b+15a^4b^2 - 20a^3b^3 +15a^2b^4- 6ab^5 + b^6$
$∴ (a + b)^6 - (a - b)^6 = 2[ 6a^5b+20a^3b^3 +6ab^5]$

Putting a = √3 and b = √2, we obtain

$(√3+ √2)^6 - (√3 −√2)^6 = 2[6(√3)^5(√2) +20 (√3)^3 (√2)3 + 6(√3)(√2)^5]$
$=2[54√6+120√6+24√6]$
$=2×198√6$
$=396√6$