Let's break down the expression:

$25^{-\frac{5}{2}}$

This expression can be rewritten using properties of exponents. First, recall that:

$a^{-b} = \frac{1}{a^b}$

So:

$25^{-\frac{5}{2}} = \frac{1}{25^{\frac{5}{2}}}$

Next, express $25$ as $5^2$, since $25 = 5^2$:

$25^{\frac{5}{2}} = (5^2)^{\frac{5}{2}} = 5^{2 \times \frac{5}{2}} = 5^5$

Thus:

$25^{-\frac{5}{2}} = \frac{1}{5^5}$

Now, calculate $5^5$:

$5^5 = 5 \times 5 \times 5 \times 5 \times 5 = 3125$

Therefore:

$25^{-\frac{5}{2}} = \frac{1}{3125}$

So, the simplified value is:

$25^{-\frac{5}{2}} = \frac{1}{3125}$

Would you like more details on exponent rules, or have any questions about this? Here's a tip:

Tip: When dealing with negative exponents, always rewrite the expression as the reciprocal of the positive exponent.

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Here are some related questions you might find interesting:

1. What is the value of $2^{-\frac{3}{4}}$?
2. How do you simplify expressions involving fractional exponents like $64^{\frac{3}{2}}$?
3. How can you evaluate $5^{-\frac{3}{4}}$ numerically?
4. What happens if the base in an expression like $25^{-x}$ is less than 1?
5. Can you simplify $\left( 16^{-\frac{3}{2}} \right)^2$?

Let me know if you want help with any of these!