1. **State the problem:** Given the equation $$(x^5)^{\frac{1}{3}} \sqrt[3]{x^2} = x^a$$ with $x > 0$, find the value of $a$. 2. **Recall the properties of exponents:** - Power of a power: $$(x^m)^n = x^{m \cdot n}$$ - Product of powers with the same base: $$x^m \cdot x^n = x^{m+n}$$ 3. **Apply the power of a power property:** $$(x^5)^{\frac{1}{3}} = x^{5 \cdot \frac{1}{3}} = x^{\frac{5}{3}}$$ 4. **Rewrite the cube root as an exponent:** $$\sqrt[3]{x^2} = x^{\frac{2}{3}}$$ 5. **Multiply the expressions using the product of powers property:** $$x^{\frac{5}{3}} \cdot x^{\frac{2}{3}} = x^{\frac{5}{3} + \frac{2}{3}} = x^{\frac{7}{3}}$$ 6. **Compare with the right side:** $$x^a = x^{\frac{7}{3}}$$ Since the bases are the same and $x > 0$, the exponents must be equal: $$a = \frac{7}{3}$$ **Final answer:** $$a = \frac{7}{3}$$