1. **State the problem:** We need to find the value of $a$ such that $$11^a = \sqrt[4]{11^3}.$$\n\n2. **Rewrite the right side using exponent rules:** The fourth root of $11^3$ can be written as an exponent: $$\sqrt[4]{11^3} = 11^{\frac{3}{4}}.$$\n\n3. **Set the exponents equal:** Since the bases are the same (11), the exponents must be equal for the equation to hold true: $$a = \frac{3}{4}.$$\n\n4. **Final answer:** $$\boxed{\frac{3}{4}}.$$