### Instruction: Simplify $
\root 3 \of {x \root 3 \of {x \root 3 \of {x \sqrt{x}}}}.
$ Express your answer in simplest radical form in terms of $x$.### Response: I see that this expression has nested roots, so I want to rewrite it using fractional exponents.
Recall that $\root n \of {x^m} = x^{m/n}$, so I can rewrite each root as a power of $x$ with a fractional exponent.
For the innermost root, I have $\sqrt{x} = x^{1/2}$.
For the next root, I have $\root 3 \of {x \sqrt{x}} = \root 3 \of {x^{3/2}} = x^{3/6} = x^{1/2}$.
For the next root, I have $\root 3 \of {x \root 3 \of {x \sqrt{x}}} = \root 3 \of {x^{3/2}} = x^{1/2}$.
For the outermost root, I have $\root 3 \of {x \root 3 \of {x \root 3 \of {x \sqrt{x}}}} = \root 3 \of {x^{3/2}} = x^{1/2}$.
So the whole expression simplifies to $x^{1/2}$.
This is already in simplest radical form, so I'm done.