1. **State the problem:** Solve for $a$ given the equation $a^{\frac{1}{8}} = \sqrt{3}$.\n\n2. **Rewrite the equation:** Note that $\sqrt{3} = 3^{\frac{1}{2}}$. So the equation becomes $$a^{\frac{1}{8}} = 3^{\frac{1}{2}}.$$\n\n3. **Raise both sides to the 8th power** to isolate $a$: $$\left(a^{\frac{1}{8}}\right)^8 = \left(3^{\frac{1}{2}}\right)^8.$$\n\n4. **Simplify the exponents:** Using the power of a power rule, $$a = 3^{\frac{1}{2} \times 8} = 3^4.$$\n\n5. **Calculate:** $$3^4 = 3 \times 3 \times 3 \times 3 = 81.$$\n\n6. **Final answer:** $$a = 81.$$