To determine the value of [tex]\(\sqrt{a^2}\)[/tex] for any real number [tex]\(a\)[/tex], let's analyze it step-by-step.
1. Understand the Expression: The given expression is [tex]\(\sqrt{a^2}\)[/tex].
2. Recall the Properties of Squares and Square Roots:
- When you square any real number [tex]\(a\)[/tex], you get [tex]\(a^2\)[/tex], which is always non-negative (i.e., [tex]\(a^2 \geq 0\)[/tex]).
- The square root function, [tex]\(\sqrt{x}\)[/tex], by definition, returns the non-negative square root of [tex]\(x\)[/tex].
3. Apply the Square Root to the Square:
- Since [tex]\(a^2\)[/tex] is always non-negative, taking the square root of [tex]\(a^2\)[/tex] gives us a non-negative number.
4. Interpret the Result:
- For any real number [tex]\(a\)[/tex], [tex]\(\sqrt{a^2}\)[/tex] simplifies to the non-negative value corresponding to [tex]\(a\)[/tex]. This is precisely the definition of the absolute value of [tex]\(a\)[/tex].
Therefore, the result of [tex]\(\sqrt{a^2}\)[/tex] is [tex]\(|a|\)[/tex].
Given the options:
A. [tex]\(a\)[/tex]
B. [tex]\(a^2\)[/tex]
C. 1
D. [tex]\(|a|\)[/tex]
The correct answer is:
[tex]\[ \boxed{|a|} \][/tex]
