The given expression is:

$\left( 5^{-3} \right)^{8x}$

We are asked to rewrite this as $5^{f(x)}$ and to find $f(x)$.

Step-by-Step Solution:

1. Apply the power of a power rule: $(a^m)^n = a^{m \cdot n}$. $\left( 5^{-3} \right)^{8x} = 5^{-3 \cdot 8x}$

2. Simplify the exponent: $5^{-3 \cdot 8x} = 5^{-24x}$

So, $5^{f(x)} = 5^{-24x}$. Therefore, $f(x) = -24x$.

Final Answer:

$f(x) = -24x$

Would you like more details on this?

Here are some follow-up questions:

1. How would you solve a similar problem with different powers, such as $(7^{-4})^{5x}$?
2. How does the power rule apply to more complex expressions involving exponents?
3. Can you express $\left( 2^{5} \right)^{3x}$ in a similar form as $2^{g(x)}$?
4. What happens when the base of the exponential expression is a fraction or a decimal?
5. How would the solution change if the initial expression had a positive exponent instead?

Tip: When dealing with exponential expressions, always simplify the exponents first to make the rewriting process easier.