To simplify the expression [tex]\( 5^{-\frac{5}{2}} \cdot 5^{\frac{9}{2}} \)[/tex], we can use the properties of exponents. Specifically, we can use the exponent rule for multiplying powers with the same base, which states:
[tex]\[ a^m \cdot a^n = a^{m+n} \][/tex]
In this case, the base [tex]\(a\)[/tex] is 5, and the exponents are [tex]\(-\frac{5}{2}\)[/tex] and [tex]\(\frac{9}{2}\)[/tex].
Step-by-step, we proceed as follows:
1. Combine the Exponents:
[tex]\[
5^{-\frac{5}{2}} \cdot 5^{\frac{9}{2}} = 5^{-\frac{5}{2} + \frac{9}{2}}
\][/tex]
2. Add the Exponents:
[tex]\[
-\frac{5}{2} + \frac{9}{2} = \frac{9}{2} - \frac{5}{2} = \frac{9 - 5}{2} = \frac{4}{2} = 2
\][/tex]
3. Simplify the Expression:
[tex]\[
5^{-\frac{5}{2} + \frac{9}{2}} = 5^2
\][/tex]
4. Calculate the Final Value:
[tex]\[
5^2 = 25
\][/tex]
Thus, the simplified value of the expression [tex]\( 5^{-\frac{5}{2}} \cdot 5^{\frac{9}{2}} \)[/tex] is [tex]\( 25 \)[/tex].
Therefore, the correct answer is:
[tex]\[
\boxed{25}
\][/tex]
