Simplify Express your answer in simplest radical form in terms of .

Simplify  Express your answer in simplest radical form in terms of .

$$\boxed{~~ \text{Formula: } \sqrt[n]{a\cdot b} = \sqrt[n]{a}\cdot \sqrt[n]{b} ~~}$$

$$\root 3 \of {x \root 3 \of {x \root 3 \of {x \sqrt{x}}}} \\\\ = \root 3 \of {x} \cdot \root 3 \of {\root 3 \of {x \root 3 \of {x \sqrt{x}}}}\\\\ = \root 3 \of {x} \cdot \root 9 \of {x \root 3 \of {x \sqrt{x}}}\\\\ = \root 3 \of {x} \cdot \root 9 \of {x} \cdot \root 9 \of {\root 3 \of {x \sqrt{x}}}\\\\ = \root 3 \of {x} \cdot \root 9 \of {x} \cdot \root 27 \of {x \sqrt{x}}\\\\ = \root 3 \of {x} \cdot \root 9 \of {x} \cdot \root 27 \of {x} \cdot \root 27 \of {\sqrt{x}}\\\\ = \root 3 \of {x} \cdot \root 9 \of {x} \cdot \root 27 \of {x} \cdot \root 54 \of {x}\\\\ = x^{\frac13} \cdot x^{\frac19} \cdot x^{\frac1{27}} \cdot x^{\frac1{54}}\\\\ = x^{\frac13+\frac19+\frac1{27}+ \frac1{54}}\\\\ = x^{\frac13 \cdot \frac{18}{18} +\frac19 \cdot \frac{6}{6} +\frac1{27} \cdot \frac{2}{2} + \frac1{54}}\\\\ =x^{\frac{18+6+2+1}{54} } \\\\$$

$$\\=x^{\frac{27}{54}}\\\\ =x^{\frac{1}{2} } \\\\ \mathbf{= \sqrt{x}}$$

√x

Basically, we're taking the cube root of x^(3/2) = x^(1/2) and multiplying this by x  = (x)^(3/2)....then taking the cube root of that, etc.........the last operation results in  (x)^(3/2)^(1/3)  = x^(1/2)  =

√x

You made that look REALLY hard Heureka   LOL   :))