To simplify and accurately explain the statement [tex]\(\sqrt{9} = 9^{\frac{1}{2}}\)[/tex], we need to follow the properties of exponents and radicals.
Let's break down the statement and simplify it step by step based on the choices provided.
### Choice 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]
This explanation uses the correct properties of exponents:
1. [tex]\( (a^m)^n = a^{mn} \)[/tex]
2. [tex]\( a^m \cdot a^n = a^{m+n} \)[/tex]
### Choice B
[tex]\[
\left(9^{\frac{1}{2}}\right)^2 = 9^{\frac{1}{2}} \cdot 9^{\frac{1}{2}} = 9 \cdot\left(\frac{1}{2}+\frac{1}{2}\right)=9 \cdot \frac{2}{2}=9 \cdot 1=9
\][/tex]
This explanation incorrectly interprets the exponentiation rules. Multiplication should be performed on the exponents rather than the base multiplied directly as shown.
### Choice C
[tex]\[
\left(9^{\frac{1}{2}}\right)^2 = 9^{\frac{1}{2}} \cdot 9^{\frac{1}{2}} = 2 \cdot 9^{\frac{1}{2}} = 2 \cdot \frac{1}{2} \cdot 9 = 1 \cdot 9 = 9
\][/tex]
This explanation improperly manipulates the terms and factors. Specifically, multiplying by 2 and then adjusting the fractional part incorrectly represents the exponent rules.
### Choice D
[tex]\[
\left(9^{\frac{1}{2}}\right)^2 = 9^{\frac{1}{2}} \cdot 9^{\frac{1}{2}} = 9^{\frac{1}{2} \cdot \frac{1}{2}} = 9^{\frac{2}{2}} = 9^1 = 9
\][/tex]
This explanation misuses the exponent rules, as the exponents should be added, not multiplied.
Based on the accurate application of exponent properties, the correct choice is:
[tex]\[
\boxed{A}
\][/tex]
