Let's solve the expression [tex]\(\left(625^2\right)^{1 / 8}\)[/tex] step-by-step.
1. First, consider the inner part of the expression: [tex]\(625^2\)[/tex].
2. We know that [tex]\(625 = 25^2\)[/tex]. Therefore, [tex]\(625^2\)[/tex] can be rewritten as:
[tex]\[
(25^2)^2
\][/tex]
3. According to the laws of exponents, [tex]\((a^m)^n = a^{mn}\)[/tex], so:
[tex]\[
(25^2)^2 = 25^{2 \cdot 2} = 25^4
\][/tex]
4. Now we need to evaluate [tex]\((25^4)^{1/8}\)[/tex].
5. Again, using the laws of exponents [tex]\((a^m)^{1/n} = a^{m/n}\)[/tex], we can rewrite the expression as:
[tex]\[
25^{4 \cdot \frac{1}{8}} = 25^{\frac{4}{8}} = 25^{\frac{1}{2}}
\][/tex]
6. The expression [tex]\(25^{\frac{1}{2}}\)[/tex] represents the square root of 25:
[tex]\[
25^{\frac{1}{2}} = \sqrt{25}
\][/tex]
7. We know that [tex]\(\sqrt{25} = 5\)[/tex].
Hence, the value of the expression [tex]\(\left(625^2\right)^{1 / 8}\)[/tex] is [tex]\(5\)[/tex].
Therefore, the correct answer is:
C. 5
