Question

what is the value of
$\sqrt{2 \sqrt{2 \sqrt{2 \sqrt{2 \sqrt{2 \sqrt{2 \sqrt{2 \sqrt{2...} } } } } } } }$
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Answer 1

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Let √2√2√2... = X

X = √2x

X ^2 = 2x

X ^2 - 2x = 0

X(X-2)=0

X = 0 Or X = 2.

Answer 2

Answer:-

This is a very easy question. Check it out.

Let us assume that,

$\sf x = \sqrt{2 \sqrt{2 \sqrt{2 \sqrt{2 \sqrt{2......} } } } }$

Squaring both sides, we get,

$\sf \implies {x}^{2} = 2 \sqrt{2 \sqrt{2 \sqrt{2 \sqrt{2....} } } }$

As mentioned earlier that,

$\sf x = \sqrt{2 \sqrt{2 \sqrt{2 \sqrt{2 \sqrt{2......} } } } }$

So,

$\sf \implies {x}^{2} = 2 \times x$

$\sf \implies {x}^{2} = 2x$

$\sf \implies {x}^{2} - 2x = 0$

$\sf \implies x(x - 2) = 0$

Therefore, either x=0 or x-2=0.

But x=0 can't be a solution of the equation.

So,

$\sf x - 2 = 0$

$\sf \implies x = 2$

Hence, x = 2 is the solution for the question.