Many square roots multiplied

Let x= (√2018(√2018(√2018...)

=> x=√2018x

=> x=2018

$\begin{aligned} x &= \sqrt{2018\sqrt{2018\sqrt{2018\sqrt{\ldots}}}} \\ x &= \sqrt{2018\underbrace{\sqrt{2018\sqrt{2018\sqrt{\ldots}}}}_{x}} \\ x &= \sqrt{2018x} \\ x^2 &= 2018x \\ x &= \boxed{2018} \quad [x \ne 0] \end{aligned}$

Let $x$ be the given number. Then

$\begin{aligned} x & = \sqrt{2018\sqrt{2018\sqrt{2018\sqrt \cdots}}} & \small \blue{\text{Squaring both sides}} \\ x^2 & = 2018x & \small \blue{\text{Since }x \ne 0} \\ \implies x & = \boxed{2018} \end{aligned}$