1. **Stating the problem:** We want to simplify the expression $$\sqrt{x^{\frac{2}{3}}}$$. 2. **Recall the property of radicals and exponents:** The square root of a number is the same as raising that number to the power of $\frac{1}{2}$. So, $$\sqrt{a} = a^{\frac{1}{2}}$$. 3. **Apply this to the given expression:** $$\sqrt{x^{\frac{2}{3}}} = \left(x^{\frac{2}{3}}\right)^{\frac{1}{2}}$$. 4. **Use the power of a power rule:** $$\left(x^{a}\right)^{b} = x^{a \times b}$$. 5. **Multiply the exponents:** $$x^{\frac{2}{3} \times \frac{1}{2}} = x^{\frac{2}{3} \times \frac{1}{2}} = x^{\frac{2 \times 1}{3 \times 2}} = x^{\frac{2}{6}}$$. 6. **Simplify the fraction:** $$x^{\frac{2}{6}} = x^{\frac{1}{3}}$$. 7. **Final answer:** $$\sqrt{x^{\frac{2}{3}}} = x^{\frac{1}{3}}$$. This means the square root of $x$ raised to the two-thirds power is the same as $x$ raised to the one-third power, which is the cube root of $x$.