To simplify the expression $(5b^{-3})^4$, we need to apply the rule of exponents $(a^m)^n = a^{m \cdot n}$ to each part of the expression.

1. Apply the exponent to the constant $5$: $(5)^4 = 625$

2. Apply the exponent to $b^{-3}$: $(b^{-3})^4 = b^{-3 \cdot 4} = b^{-12}$

So, putting it all together, we have: $(5b^{-3})^4 = 625b^{-12}$

This is the simplified form of the expression. If you want to express this in a positive exponent, you can rewrite it as: $\frac{625}{b^{12}}$

Would you like further explanation or have any questions? Here are a few related questions to explore:

1. What is the general rule for simplifying powers of products like $(ab)^n$?
2. How do you simplify expressions with negative exponents?
3. How do you handle expressions with fractional exponents, such as $b^{\frac{1}{2}}$?
4. What are the properties of exponents when dividing terms with the same base?
5. Can you simplify $(3x^2y^{-3})^3$?

Tip: When you raise a power to another power, always multiply the exponents!