Evaluate

We know that-

Hence

= a⁶ + 6a⁵b + 15a⁴b² + 20a³b³ + 15a²b⁴ + 6ab⁵ + b⁶

Thus, (a + b)⁶ = a⁶ + 6a⁵b + 15a⁴b² + 20a³b³ + 15a²b⁴ + 6ab⁵ + b⁶

Replacing b with -b

(a +(-b))⁶ = a⁶ + 6a⁵(-b) + 15a⁴(-b)² + 20a³(-b)³ + 15a²(-b)⁴ + 6a(-b)⁵ + (-b)⁶

(a - b)⁶ = a⁶ - 6a⁵b + 15a⁴b² - 20a³b³ + 15a²b⁴ - 6ab⁵ + b⁶

Now,

(a + b)⁶ - (a - b)⁶

= (a⁶ + 6a⁵b + 15a⁴b² + 20a³b³ + 15a²b⁴ + 6ab⁵ + b⁶)

- (a⁶ - 6a⁵b + 15a⁴b² - 20a³b³ + 15a²b⁴ - 6ab⁵ + b⁶)

= 2(6a⁵b + 20a³b³ + 6ab⁵)

Putting a = √3 & b = √2, we get-

(√3 + √2)⁶ - (√3 - √2)⁶

= 2[6(√3)⁵(√2) + 20(√3)³(√2)³ + 6(√3)(√2)⁵]

= 2[6(9√3)(√2) + 20(3√3)(2√2) + 6(√3)(4√2)]

= 2[54√6 + 120√6 + 24√6]

= 2[198√6]

= 396√6