To solve the expression [tex]\(\left(625^2\right)^{1 / 8}\)[/tex], follow these steps:
1. Simplify the exponent:
[tex]\[
\left(625^2\right)^{1 / 8}
\][/tex]
is equivalent to
[tex]\[
\left(625^{2}\right)^{\frac{1}{8}} = 625^{2 \cdot \frac{1}{8}} = 625^{\frac{2}{8}} = 625^{\frac{1}{4}}
\][/tex]
2. Interpret the exponent [tex]\(\frac{1}{4}\)[/tex]:
The expression [tex]\(625^{\frac{1}{4}}\)[/tex] represents the fourth root of 625.
3. Evaluate [tex]\(\sqrt[4]{625}\)[/tex]:
We need to find a number [tex]\(x\)[/tex] such that [tex]\(x^4 = 625\)[/tex]. By knowing common powers, we recognize that:
[tex]\[
5^4 = 5 \times 5 \times 5 \times 5 = 25 \times 25 = 625
\][/tex]
Therefore:
[tex]\[
\sqrt[4]{625} = 5
\][/tex]
So, the value of [tex]\(\left(625^2\right)^{1 / 8}\)[/tex] is [tex]\(\boxed{5}\)[/tex].
