If $ \sqrt{x\cdot\!\sqrt{x\cdot\!\sqrt{x\cdot\!\sqrt{x\cdots}}}}=3$, find $x$.
I notice that the expression on the left is an infinite nested radical, which means that it can be written as $ \sqrt{x\cdot\!y}$, where $y$ is the same expression. So I can write $ \sqrt{x\cdot\!\sqrt{x\cdot\!\sqrt{x\cdot\!\sqrt{x\cdots}}}}=\sqrt{x\cdot\!y}$. Squaring both sides, I get $ x\cdot\!\sqrt{x\cdot\!\sqrt{x\cdot\!\sqrt{x\cdots}}}=x\cdot\!y$. Squaring both sides again, I get $ x\cdot\!\sqrt{x\cdot\!\sqrt{x\cdot\!\sqrt{x\cdots}}}=9$. Dividing both sides by $ \sqrt{x\cdot\!\sqrt{x\cdot\!\sqrt{x\cdot\!\sqrt{x\cdots}}}}$, I get $ x=\frac{9}{3}=3$. 