Solve $x\sqrt{x\sqrt{x\sqrt{x\dots}}} = 4$

sequences-and-series nested-radicals radicals

Today I faced a strange equation and I didn't manage to find a solution to it: $$x\sqrt{x\sqrt{x\sqrt{x\dots}}} = 4$$ Maybe someone will help me to find a way to solve it. By the way, this equation is from high school course.


$$x\sqrt{x\sqrt{x\sqrt{x\cdots}}}=x\,x^{1/2}\,x^{1/4}\,x^{1/8}\cdots=x^{1+1/2+1/4+1/8+\cdots}=x^2$$


Hint

Divide both sides by $x$ and square them. You should notice something beautiful.

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