Question

$\sqrt{12 \div \sqrt{12 \div \sqrt{12....} } }$
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Answer 1

Let x = $\sf{\sqrt{12 \div x} }$

Squaring both sides,

$\sf{x^2=(\sqrt{12 \div x})^2}\\\\\sf{\dashrightarrow x^2=12 \div x}\\\\\sf{\dashrightarrow x^3=12}\\\\\sf{\dashrightarrow x=\sqrt[3]{12} }$

Putting the value of x :-

$\sf{\dashrightarrow \sqrt[3]{12}=\sqrt{12 \div x} }\\\\\sf{\dashrightarrow 12\sqrt{12}=12 \div x }\\\\\sf{\dashrightarrow \sqrt{12}x=12 \div 12}\\\\\sf{\dashrightarrow x=\dfrac{1}{\sqrt{12}} }$

Putting √12 = 3.46, we get :-

x = 0.289