To solve the given expression [tex]\(\left(\frac{1}{5}\right)^{-2}\)[/tex]:
1. Understanding Negative Exponents:
By the rules of exponents, a negative exponent [tex]\(a^{-n}\)[/tex] can be rewritten as [tex]\(\frac{1}{a^n}\)[/tex].
Therefore, [tex]\(\left(\frac{1}{5}\right)^{-2}\)[/tex] can be transformed using this rule.
2. Rewriting the Expression:
Applying the rule:
[tex]\[
\left(\frac{1}{5}\right)^{-2} = \frac{1}{\left(\frac{1}{5}\right)^2} = \frac{1}{\frac{1}{25}}
\][/tex]
3. Simplifying the Fraction:
[tex]\(\frac{1}{\frac{1}{25}}\)[/tex] is the reciprocal of [tex]\(\frac{1}{25}\)[/tex].
The reciprocal of [tex]\(\frac{1}{25}\)[/tex] is 25.
Therefore:
[tex]\[
\frac{1}{\frac{1}{25}} = 25
\][/tex]
Thus, the value of the expression [tex]\(\left(\frac{1}{5}\right)^{-2}\)[/tex] is [tex]\(25\)[/tex].
So, the correct number to fill in the squares is:
[tex]\[ \boxed{25} \][/tex]
The possible answers given [tex]\(2^{\frac{1}{8}}, \frac{5}{2}, \frac{1}{9}\)[/tex] are incorrect as they do not match the correct simplified result [tex]\(25\)[/tex]. The correct answer from the provided choices would be not listed, so you should manually input [tex]\(25\)[/tex] as the result.
