To solve the equation [tex]\(\sqrt{x} = 9\)[/tex], we need to determine which values of [tex]\(x\)[/tex], if any, satisfy this equation. The square root function [tex]\(\sqrt{x}\)[/tex] is defined only for non-negative values of [tex]\(x\)[/tex]. Therefore, any negative options must be discarded immediately as they are not in the domain of the square root function.
Let's analyze each option step-by-step:
A. [tex]\(0\)[/tex]
[tex]\[
\sqrt{0} = 0 \neq 9
\][/tex]
B. [tex]\(-3\)[/tex]
The square root function is not defined for negative numbers in the real number system.
C. [tex]\(81\)[/tex]
[tex]\[
\sqrt{81} = 9
\][/tex]
D. [tex]\(3\)[/tex]
[tex]\[
\sqrt{3} \approx 1.732 \neq 9
\][/tex]
E. [tex]\(-81\)[/tex]
Again, the square root function is not defined for negative numbers in the real number system.
F. None
Given that we have already found a valid solution, this option does not apply.
From the above steps, we can see that the only value that satisfies the equation [tex]\(\sqrt{x} = 9\)[/tex] is option C:
[tex]\[
x = 81
\][/tex]
Therefore, the correct answer is:
C. 81
