To solve the expression [tex]\(\left(7^{1 / 2}\right)^2\)[/tex], let's work through the steps methodically.
1. Understand the Expression: We start with [tex]\(\left(7^{1 / 2}\right)^2\)[/tex]. The notation [tex]\(7^{1 / 2}\)[/tex] represents the square root of 7.
2. Simplify the Exponent: The expression [tex]\(\left(7^{1/2}\right)^2\)[/tex] involves raising a power to another power. We know from the properties of exponents that:
[tex]\[
\left(a^{m}\right)^n = a^{m \cdot n}
\][/tex]
3. Apply the Property: In our expression, [tex]\(a = 7\)[/tex], [tex]\(m = \frac{1}{2}\)[/tex], and [tex]\(n = 2\)[/tex]. Applying the property, we have:
[tex]\[
\left(7^{1/2}\right)^2 = 7^{(1/2) \cdot 2}
\][/tex]
4. Simplify the Exponent Computation: Compute the exponent [tex]\((1/2) \cdot 2\)[/tex]:
[tex]\[
(1/2) \cdot 2 = 1
\][/tex]
5. Final Simplification: So the expression simplifies to:
[tex]\[
7^1 = 7
\][/tex]
Therefore, the value of the expression [tex]\(\left(7^{1/2}\right)^2\)[/tex] is [tex]\(7\)[/tex].
The correct answer is [tex]\( \boxed{7} \)[/tex].
