Simplify the following expression.
$5^{-\frac{5}{2}} \cdot 5^{\frac{9}{2}}$
We are tasked with simplifying the expression \(5^{-\frac{5}{2}} \cdot 5^{\frac{9}{2}}\). Both terms share the same base, which is 5. We can apply the exponent rule for multiplication, which states that for any base \(a\) and exponents \(m\) and \(n\), the product \(a^m \cdot a^n = a^{m+n}\).
To simplify the expression, we sum the exponents:
\[-\frac{5}{2} + \frac{9}{2} = \frac{-5 + 9}{2} = \frac{4}{2} = 2\]
Thus, the expression simplifies to \(5^2\).
Now that we have simplified the expression to \(5^2\), we can evaluate it:
\[5^2 = 5 \times 5 = 25\]
Comparing this result to the given options, we see that 25 is one of the choices. Therefore, the simplified value is \(\boxed{25}\).
Exponents are essential in various fields, such as calculating compound interest in finance, determining population growth in biology, or measuring sound intensity in physics. For example, if a bacteria culture doubles every hour, its population after \(n\) hours can be modeled by an exponential growth function. Understanding how to manipulate and simplify exponents, as demonstrated here, provides a foundation for solving more complex problems in these fields efficiently.