Find (x + 1)6 + (x – 1)6. Hence, or otherwise evaluate (√2 + 1)6 + (√2 – 1)6.
The Expression can be written as
(x + 1)6 + (x - 1)6 = 2[6C0x6 + 6C2x4 + 6C3x2 + 6C6x0]
= 2[x6 + 15x4 + 15x2 + 1]
Now, Putting x = √2 we get:
(√2 + 1)6 + (√2 – 1)6= 
= 2[8 + 15 × 4 + 15 × 2 + 1]
= 2[8 + 60 + 30 + 1]
= 198
Hence, (√2 + 1)6 + (√2 – 1)6= 198
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