1. **State the problem:** We need to find the value of $a$ in the equation $$\left(x^{\frac{1}{2}}\right)^3 \sqrt{x} = x^a$$ given that $x > 0$. 2. **Recall exponent rules:** - Power of a power: $\left(x^m\right)^n = x^{mn}$ - Product of powers: $x^m \cdot x^n = x^{m+n}$ 3. **Apply the power of a power rule:** $$\left(x^{\frac{1}{2}}\right)^3 = x^{\frac{1}{2} \times 3} = x^{\frac{3}{2}}$$ 4. **Rewrite the square root:** $$\sqrt{x} = x^{\frac{1}{2}}$$ 5. **Substitute back into the original equation:** $$x^{\frac{3}{2}} \cdot x^{\frac{1}{2}} = x^a$$ 6. **Use the product of powers rule:** $$x^{\frac{3}{2} + \frac{1}{2}} = x^a$$ 7. **Add the exponents:** $$\frac{3}{2} + \frac{1}{2} = \frac{4}{2} = 2$$ 8. **Therefore:** $$x^2 = x^a$$ 9. **Since bases are equal and $x > 0$, exponents must be equal:** $$a = 2$$ **Final answer:** $a = 2$