1. The problem is to simplify and verify the expression $\left(9^{1/2}\right)^2$ and identify the correct simplification among the options. 2. Recall the exponent rule: $\left(a^m\right)^n = a^{m \cdot n}$ and the product rule: $a^m \cdot a^n = a^{m+n}$. 3. Start with the expression: $$\left(9^{1/2}\right)^2 = 9^{1/2 \cdot 2}$$ 4. Simplify the exponent multiplication: $$1/2 \cdot 2 = 1$$ 5. So, $$9^{1} = 9$$ 6. Check the options: - Option A incorrectly uses $9^{1/2 \cdot 1/2}$ instead of $9^{1/2 \cdot 2}$. - Option B incorrectly treats multiplication of powers as multiplication by 2. - Option C correctly applies the product rule for powers: $9^{1/2} \cdot 9^{1/2} = 9^{1/2 + 1/2} = 9^{1} = 9$. - Option D incorrectly multiplies the base by the sum of exponents. 7. Therefore, the correct simplification is option C. Final answer: $\boxed{9}$