1. **State the problem:** We need to express the equation $8\sqrt{a} = a^n$ in terms of $n$. 2. **Recall the rules:** The square root of $a$ can be written as a fractional exponent: $\sqrt{a} = a^{\frac{1}{2}}$. 3. **Rewrite the equation:** Substitute $\sqrt{a}$ with $a^{\frac{1}{2}}$: $$8 a^{\frac{1}{2}} = a^n$$ 4. **Express 8 as a power of $a$ if possible:** Since 8 is a constant and $a$ is a variable, we cannot rewrite 8 as a power of $a$ unless $a$ is a specific number. So, we isolate $a^n$: $$a^n = 8 a^{\frac{1}{2}}$$ 5. **Divide both sides by $a^{\frac{1}{2}}$ to isolate $a^{n - \frac{1}{2}}$:** $$\frac{a^n}{a^{\frac{1}{2}}} = 8$$ 6. **Use the exponent division rule:** $$a^{n - \frac{1}{2}} = 8$$ 7. **Take the logarithm base $a$ of both sides to solve for $n$:** $$n - \frac{1}{2} = \log_a 8$$ 8. **Solve for $n$:** $$n = \log_a 8 + \frac{1}{2}$$ **Final answer:** $$n = \log_a 8 + \frac{1}{2}$$