1. **State the problem:** Simplify the radical expression $$\sqrt{a^7}$$. 2. **Recall the rule for radicals and exponents:** $$\sqrt{a^n} = a^{\frac{n}{2}}$$. 3. **Apply the rule:** $$\sqrt{a^7} = a^{\frac{7}{2}}$$ 4. **Rewrite the exponent as a sum of an integer and a fraction:** $$a^{\frac{7}{2}} = a^{3 + \frac{1}{2}} = a^3 \cdot a^{\frac{1}{2}}$$ 5. **Express the fractional exponent as a radical:** $$a^3 \cdot a^{\frac{1}{2}} = a^3 \cdot \sqrt{a}$$ 6. **Final simplified form:** $$\sqrt{a^7} = a^3 \sqrt{a}$$ **Answer:** The unknown exponent in the expression $$a^{?} \sqrt{a}$$ is 3.