What is value of $\sqrt[n]{x\sqrt[n]{x\sqrt[n]{x\sqrt[n]{x\ldots}}}}=$? My question: how to find nested radical having $n$th roots $$\large\sqrt[n]{x\sqrt[n]{x\sqrt[n]{x\sqrt[n]{x\ldots}}}}=?$$
My try:
$$\large\sqrt[n]{x\sqrt[n]{x\sqrt[n]{x\sqrt[n]{x\ldots}}}}=y$$
$$\large\sqrt[n]{xy}=y$$
$$\large xy=y^n$$$$ y^n-xy=0$$
Above $\mathrm{nth}$ degree equation should $n$ number of roots but my teacher says it has a unique value. I am totally confused how to get that value .  
What will be the answer? I am looking for some trick or elegant way to solve such nested radical. Thanks  

Here is a trick:
$$\large \sqrt[n]{x\sqrt[n]{x\sqrt[n]{x\ldots}}}=x^{\frac{1}{n}}\cdot x^{\frac{1}{n^2}}\cdot x^{\frac{1}{n^3}}\cdot\ldots=x^{\left(\frac{1}{n}+\frac{1}{n^2}+\frac{1}{n^3}+\ldots\right)}=x^{\frac{\frac1n}{1-\frac1n}}=x^{\frac{1}{n-1}} $$
