To solve the expression [tex]\(\sqrt{a^2}\)[/tex] for any real number [tex]\(a\)[/tex], let's proceed through the following steps:
1. Understand the Expression:
[tex]\[
\sqrt{a^2}
\][/tex]
2. Consider the Definition of Square Root and Squaring:
The expression [tex]\(a^2\)[/tex] represents [tex]\(a\)[/tex] squared, meaning [tex]\(a \times a\)[/tex]. The square root function undoes squaring. Thus, when we take the square root of [tex]\(a^2\)[/tex], we are looking for a value that, when squared, gives us [tex]\(a^2\)[/tex].
3. Absolute Value Insight:
For any real number [tex]\(a\)[/tex], whether [tex]\(a\)[/tex] is positive or negative, squaring it will always yield a non-negative value:
- If [tex]\(a \geq 0\)[/tex], then [tex]\(a^2 \geq 0\)[/tex]
- If [tex]\(a < 0\)[/tex], then also [tex]\(a^2 > 0\)[/tex]
4. Square Root of a Square:
The square root of a squared value, [tex]\(\sqrt{a^2}\)[/tex], results in the non-negative value that was squared in the first place. This is essentially the absolute value of [tex]\(a\)[/tex], denoted as [tex]\(|a|\)[/tex]:
[tex]\[
\sqrt{a^2} = |a|
\][/tex]
5. Conclusion:
Therefore, [tex]\(\sqrt{a^2} = |a|\)[/tex] for any real number [tex]\(a\)[/tex].
The correct choice is:
[tex]\[
\boxed{|a|}
\][/tex]
