Let's break down the given expression step by step:

The expression is:

$\left( \frac{\sqrt[5]{\frac{x^{25}}{3125}}}{\ } \right)^{-3} \times \frac{1}{x^{-15}}$

Step 1: Simplify the fifth root

We need to simplify the part inside the parentheses first, which is the fifth root of $\frac{x^{25}}{3125}$. Using the rule for roots and exponents:

[ \sqrt[5]{\frac{x^{25}}{3125}} = \frac{\sqrt[5]{x^{25}}}{\sqrt[5]{3125}} = \frac{x^{5}}{5} ]

Since $\sqrt[5]{x^{25}} = x^5$ and $\sqrt[5]{3125} = 5$ (since $3125 = 5^5$).

Now the expression becomes:

$\left( \frac{x^5}{5} \right)^{-3} \times \frac{1}{x^{-15}}$

Step 2: Apply the negative exponent

Next, we apply the $-3$ exponent to the fraction $\frac{x^5}{5}$:

$\left( \frac{x^5}{5} \right)^{-3} = \frac{5^3}{x^{15}} = \frac{125}{x^{15}}$

So now, the entire expression becomes:

$\frac{125}{x^{15}} \times \frac{1}{x^{-15}}$

Step 3: Simplify the multiplication

We now multiply the two terms:

$\frac{125}{x^{15}} \times \frac{1}{x^{-15}} = \frac{125}{x^{15} \times x^{-15}} = \frac{125}{x^{0}} = 125$

Thus, the expression simplifies to:

$125$

Final Answer:

The simplified value of the given expression is $125$.

Do you have any questions or would you like a deeper explanation of any of these steps?

Here are five related questions to explore further:

1. How do we simplify expressions involving roots and exponents?
2. What are the rules for handling negative exponents?
3. How do we apply exponents to both the numerator and denominator of a fraction?
4. Can you give another example where a negative exponent is applied?
5. What happens if the base of the exponent is negative?

Tip: When simplifying expressions with exponents, always handle the root and exponent first before applying any multiplications or divisions.