1. **State the problem:** Simplify the expression $$\frac{\left(x^{-\frac{1}{3}} \cdot \sqrt[3]{x^2}\right)^3}{x^{-2}}$$. 2. **Recall the rules:** - When multiplying powers with the same base, add exponents: $$a^m \cdot a^n = a^{m+n}$$. - When raising a power to another power, multiply exponents: $$(a^m)^n = a^{m \cdot n}$$. - Division of powers with the same base subtracts exponents: $$\frac{a^m}{a^n} = a^{m-n}$$. - The cube root can be written as a fractional exponent: $$\sqrt[3]{x^2} = x^{\frac{2}{3}}$$. 3. **Rewrite the expression inside the parentheses:** $$x^{-\frac{1}{3}} \cdot x^{\frac{2}{3}} = x^{-\frac{1}{3} + \frac{2}{3}} = x^{\frac{1}{3}}$$. 4. **Raise the result to the power 3:** $$\left(x^{\frac{1}{3}}\right)^3 = x^{\frac{1}{3} \cdot 3} = x^1 = x$$. 5. **Divide by $$x^{-2}$$:** $$\frac{x}{x^{-2}} = x^{1 - (-2)} = x^{1 + 2} = x^3$$. 6. **Final answer:** $$x^3$$.