Firstly, let us solve the given expression:
(x + 1) 6 + (x – 1) 6
The above expression can be expressed as,
(x + 1) 6 + (x – 1) 6 = 2 [6C0 x6 + 6C2 x4 + 6C4 x2 + 6C6 x0]
= 2 [x6 + 15x4 + 15x2 + 1]
Now,
Let us evaluate the expression:
\( (\sqrt{2} + 1)^6 + (\sqrt{2} – 1)^6\)
So consider, x = \( \sqrt{2}\) then we get,
\( (\sqrt{2} + 1)^6 + (\sqrt{2} – 1)^6\)= 2 [x6 + 15x4 + 15x2 + 1]
= 2 [(\( \sqrt{2}\))6 + 15 (\( \sqrt{2}\))4 + 15 (\( \sqrt{2}\))2 + 1]
= 2 [8 + 15 (4) + 15 (2) + 1]
= 2 [8 + 60 + 30 + 1]
= 198
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