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jmc

algebra intermediate

Problem

Simplify

\root 3 \of x \root 3 \of x \root 3 \of x x .

Express your answer in simplest radical form in terms of

x

.

Solution

We have

\root 3 \of x \root 3 \of x \root 3 \of x x = (x (x (x \cdot x^{\frac{1}{2}})^{\frac{1}{3}})^{\frac{1}{3}})^{\frac{1}{3}} = (x (x (x^{\frac{3}{2}})^{\frac{1}{3}})^{\frac{1}{3}})^{\frac{1}{3}} = (x (x \cdot x^{\frac{1}{2}})^{\frac{1}{3}})^{\frac{1}{3}} = (x (x^{\frac{3}{2}})^{\frac{1}{3}})^{\frac{1}{3}} = (x \cdot x^{\frac{1}{2}})^{\frac{1}{3}} = (x^{\frac{3}{2}})^{\frac{1}{3}} = x^{\frac{1}{2}} = x .

Final answer

\sqrt{x}