1. **State the problem:** Simplify the expression $$(x^{\frac{1}{2}})^7 \times \sqrt{x^9}$$. 2. **Recall the rules:** - Power of a power: $$(a^m)^n = a^{mn}$$ - Square root as exponent: $$\sqrt{x} = x^{\frac{1}{2}}$$ - Multiply powers with the same base: $$a^m \times a^n = a^{m+n}$$ 3. **Apply the power of a power rule:** $$(x^{\frac{1}{2}})^7 = x^{\frac{1}{2} \times 7} = x^{\frac{7}{2}}$$ 4. **Rewrite the square root:** $$\sqrt{x^9} = (x^9)^{\frac{1}{2}} = x^{9 \times \frac{1}{2}} = x^{\frac{9}{2}}$$ 5. **Multiply the expressions:** $$x^{\frac{7}{2}} \times x^{\frac{9}{2}} = x^{\frac{7}{2} + \frac{9}{2}} = x^{\frac{16}{2}} = x^8$$ 6. **Final answer:** $$x^8$$