### Instruction: If $ \sqrt{x\cdot\!\sqrt{x\cdot\!\sqrt{x\cdot\!\sqrt{x\cdots}}}}=3$, find $x$.### Response: 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$.
Dividing both sides by $x$, I get $ \sqrt{x\cdot\!\sqrt{x\cdot\!\sqrt{x\cdot\!\sqrt{x\cdots}}}}=y$.
But this is the same as the original equation, so I can substitute $y=3$.
Therefore, I have $ \sqrt{x\cdot\!\sqrt{x\cdot\!\sqrt{x\cdot\!\sqrt{x\cdots}}}}=3$.
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$.