To solve the expression [tex]\(\left(8^{1 / 2}\right)^2\)[/tex], we need to understand a few fundamental concepts about exponents.
1. Exponentiation and Roots: The expression [tex]\(8^{1 / 2}\)[/tex] represents the square root of 8. In other words, [tex]\(8^{1 / 2}\)[/tex] is the number which, when squared, gives 8.
2. Simplifying the Expression: When we raise [tex]\(8^{1 / 2}\)[/tex] to the power of 2, we are essentially squaring the square root of 8. Mathematically, this looks like:
[tex]\[
\left(8^{1 / 2}\right)^2 = (8^{1 / 2}) \cdot (8^{1 / 2})
\][/tex]
According to the properties of exponents, [tex]\((a^{m})^{n} = a^{m \cdot n}\)[/tex]. In our case:
[tex]\[
\left(8^{1 / 2}\right)^2 = 8^{(1/2) \cdot 2}
\][/tex]
3. Simplification of Exponents: Now, we multiply the exponents:
[tex]\[
(1 / 2) \cdot 2 = 1
\][/tex]
Thus, the expression simplifies to:
[tex]\[
8^{1} = 8
\][/tex]
The exact value of the expression [tex]\(\left(8^{1 / 2}\right)^2\)[/tex] is [tex]\(8\)[/tex].
Therefore, the correct answer is:
C. 8
