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Question Number 8362 by Nayon last updated on 09/Oct/16

\sqrt{2 \sqrt{2 \sqrt{2 \sqrt{2 \sqrt{2 \sqrt{2 \dots . .}}}}}} = ?

Answered by Rasheed Soomro last updated on 09/Oct/16

Let x = \sqrt{2 \sqrt{2 \sqrt{2 \sqrt{2 \sqrt{2 \sqrt{2 \dots . .}}}}}}

x^{2} = 2 \sqrt{2 \sqrt{2 \sqrt{2 \sqrt{2 \sqrt{2 \dots . .}}}}}

x^{2} = 2 x

x^{2} - 2 x = 0

x (x - 2) = 0

x = 0 ∣ x - 2 = 0 \Rightarrow x = 2

Obviously x \neq 0, so x = 2

Hence

\sqrt{2 \sqrt{2 \sqrt{2 \sqrt{2 \sqrt{2 \sqrt{2 \dots . .}}}}}} = 2