Let's explore the question step-by-step to determine the value of [tex]\((\sqrt{c})^2\)[/tex] for any nonnegative real number [tex]\(c\)[/tex].
1. Understand the Square Root Operation: By definition, the square root of a number [tex]\( c \)[/tex] (denoted as [tex]\(\sqrt{c}\)[/tex]) is a number that, when multiplied by itself, gives [tex]\( c \)[/tex]. Mathematically, [tex]\(\sqrt{c}\)[/tex] is such that:
[tex]\[
(\sqrt{c}) \times (\sqrt{c}) = c
\][/tex]
2. Squaring the Square Root: We need to evaluate [tex]\((\sqrt{c})^2\)[/tex]. When you square the square root of a number, you are essentially reversing the square root operation. That means:
[tex]\[
(\sqrt{c})^2 = \sqrt{c} \times \sqrt{c}
\][/tex]
3. Simplification: Based on the definition of the square root operation, multiplying [tex]\(\sqrt{c}\)[/tex] by itself will just return the original number [tex]\( c \)[/tex]:
[tex]\[
\sqrt{c} \times \sqrt{c} = c
\][/tex]
4. Conclusion: Therefore, the expression [tex]\((\sqrt{c})^2\)[/tex] simplifies to:
[tex]\[
(\sqrt{c})^2 = c
\][/tex]
Hence, for any nonnegative real number [tex]\( c \)[/tex],
[tex]\[
(\sqrt{c})^2 = c
\][/tex]
