1. **State the problem:** Simplify the expression $$(x^5)^{\frac{1}{3}} \cdot \sqrt[3]{x^2}$$ and express it in the form $x^a$. 2. **Recall the exponent rules:** - Power of a power: $$(x^m)^n = x^{m \cdot n}$$ - Product of powers with the same base: $$x^m \cdot x^n = x^{m+n}$$ - Cube root as fractional exponent: $$\sqrt[3]{x^k} = x^{\frac{k}{3}}$$ 3. **Apply the power of a power rule:** $$(x^5)^{\frac{1}{3}} = x^{5 \cdot \frac{1}{3}} = x^{\frac{5}{3}}$$ 4. **Rewrite the cube root:** $$\sqrt[3]{x^2} = x^{\frac{2}{3}}$$ 5. **Multiply the expressions using product of powers:** $$x^{\frac{5}{3}} \cdot x^{\frac{2}{3}} = x^{\frac{5}{3} + \frac{2}{3}} = x^{\frac{7}{3}}$$ 6. **Final answer:** $$x^a = x^{\frac{7}{3}}$$ Therefore, $$a = \frac{7}{3}$$.