Find (x + 1) 6 + (x – 1) 6. Hence, or otherwise evaluate $$(\sqrt{2} + 1)^6 + (\sqrt{2} – 1)^6$$

Asked by Aaryan | 1 year ago |  110

##### Solution :-

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

Answered by Aaryan | 1 year ago

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