1. **Problem Statement:** Find the value of $a$ if $a^{\sqrt{a}} = (a \sqrt{a})^{a}$. 2. **Formula and Rules:** We use properties of exponents: $$x^{m} = y^{n} \implies \text{if bases are equal, then } m = n.$$ Also, $a \sqrt{a} = a \cdot a^{\frac{1}{2}} = a^{\frac{3}{2}}$. 3. **Rewrite the equation:** $$a^{\sqrt{a}} = (a \sqrt{a})^{a} = \left(a^{\frac{3}{2}}\right)^{a} = a^{\frac{3a}{2}}.$$ 4. **Equate exponents (since bases are equal and $a>0$):** $$\sqrt{a} = \frac{3a}{2}.$$ 5. **Solve for $a$: Let $\sqrt{a} = t$, so $a = t^{2}$. Substitute:** $$t = \frac{3 t^{2}}{2} \implies 2t = 3 t^{2} \implies 3 t^{2} - 2 t = 0 \implies t(3 t - 2) = 0.$$ 6. **Solutions for $t$:** $$t = 0 \quad \text{or} \quad t = \frac{2}{3}.$$ 7. **Since $t = \sqrt{a} > 0$, discard $t=0$. So:** $$\sqrt{a} = \frac{2}{3} \implies a = \left(\frac{2}{3}\right)^{2} = \frac{4}{9}.$$ **Final answer:** $$a = \frac{4}{9}.$$