To solve for [tex]\(\sqrt{c^2}\)[/tex] for any real number [tex]\(c\)[/tex], let's carefully analyze the expression.
1. Understand the Meaning:
- The expression [tex]\(c^2\)[/tex] represents [tex]\(c\)[/tex] squared, which means [tex]\(c\)[/tex] multiplied by itself.
- The square root function, denoted [tex]\(\sqrt{\ \ }\)[/tex], returns the principal (non-negative) square root of its argument.
2. Simplify the Expression:
- Since [tex]\(c^2\)[/tex] is always non-negative (as any real number squared is non-negative), we are looking for the non-negative number that, when squared, results in [tex]\(c^2\)[/tex].
3. Square Root and Absolute Value Relationship:
- The square root of [tex]\(c^2\)[/tex] gives a non-negative result, which corresponds to the absolute value of [tex]\(c\)[/tex].
- This can be understood as follows: [tex]\(\sqrt{c^2} = |c|\)[/tex] because the absolute value of [tex]\(c\)[/tex] is defined as:
[tex]\[
|c| = \begin{cases}
c & \text{if } c \geq 0 \\
-c & \text{if } c < 0
\end{cases}
\][/tex]
- For any [tex]\(c \geq 0\)[/tex], [tex]\(\sqrt{c^2} = c\)[/tex].
- For any [tex]\(c < 0\)[/tex], [tex]\(\sqrt{c^2} = -c\)[/tex] (which is positive).
So, the correct answer is [tex]\(|c|\)[/tex].
Thus,
[tex]\[
\sqrt{c^2} = |c|
\][/tex]
The answer is [tex]\(\boxed{|c|}\)[/tex], which corresponds to option C.
