Sure! Let's go through each option step-by-step to find the correct simplification that accurately explains the statement [tex]\(\sqrt{9} = 9^{\frac{1}{2}}\)[/tex].
### Option A
1. Start with [tex]\(\left(9^{\frac{1}{2}}\right)^2\)[/tex].
2. Rewrite as [tex]\(9^{\frac{1}{2}} \cdot 9^{\frac{1}{2}}\)[/tex].
3. Apply the exponent addition rule: [tex]\(9^{\frac{1}{2} + \frac{1}{2}}\)[/tex].
4. Simplify the exponent: [tex]\(9^{\frac{2}{2}}\)[/tex].
5. Simplify the fraction: [tex]\(9^1\)[/tex].
6. Result: [tex]\(9\)[/tex].
### Option B
1. Start with [tex]\(\left(9^{\frac{1}{2}}\right)^2\)[/tex].
2. Rewrite as [tex]\(9^{\frac{1}{2}} \cdot 9^{\frac{1}{2}}\)[/tex].
3. Incorrectly compute as [tex]\(9 \cdot \left(\frac{1}{2} + \frac{1}{2}\)[/tex]\), which should not be done since the base [tex]\(9\)[/tex] is constant.
4. Proceed with incorrect steps leading to: [tex]\(9 \cdot \frac{2}{2}\)[/tex].
5. Simplify to: [tex]\(9 \cdot 1 = 9\)[/tex].
While the final result is correct, the steps contain an error. Exponent addition should not be translated into multiplication with the base.
### Option C
1. Start with [tex]\(\left(9^{\frac{1}{2}}\right)^2\)[/tex].
2. Rewrite as [tex]\(9^{\frac{1}{2}} \cdot 9^{\frac{1}{2}}\)[/tex].
3. Apply the exponent addition rule: [tex]\(9^{\frac{1}{2} + \frac{1}{2}}\)[/tex].
4. Simplify the exponent: [tex]\(9^{\frac{2}{2}}\)[/tex].
5. Simplify the fraction: [tex]\(9^1\)[/tex].
6. Result: [tex]\(9\)[/tex].
### Option D
1. Start with [tex]\(\left(9^{\frac{1}{2}}\right)^2\)[/tex].
2. Rewrite as [tex]\(9^{\frac{1}{2}} \cdot 9^{\frac{1}{2}}\)[/tex].
3. Incorrectly compute as [tex]\(2 \cdot 9^{\frac{1}{2}}\)[/tex] which introduces an incorrect multiplication factor.
4. Proceed with erroneous steps resulting in incorrect intermediate calculations.
5. Reaches [tex]\(9\)[/tex] erroneously.
### Conclusion
The correct options are those which accurately follow mathematical rules for exponents and correctly apply arithmetic operations. Here, these are:
- Option A: [tex]\(\left(9^{\frac{1}{2}}\right)^2 = 9^{\frac{1}{2}} \cdot 9^{\frac{1}{2}} = 9^{\frac{1}{2} + \frac{1}{2}} = 9^{\frac{2}{2}} = 9^1 = 9\)[/tex]
- Option C: [tex]\(\left(9^{\frac{1}{2}}\right)^2 = 9^{\frac{1}{2}} \cdot 9^{\frac{1}{2}} = 9^{\frac{1}{2} + \frac{1}{2}} = 9^{\frac{2}{2}} = 9^1 = 9\)[/tex]
Since we are to select one correct option, either A or C would be correct. With the given approach, we choose:
Option A
