Answer :

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