We are asked to simplify the following expression:

$\frac{ \left( -3b^4 \right)^2 }{-12b^{-9}}$

Step 1: Simplify the numerator

In the numerator, $\left( -3b^4 \right)^2$, we apply the power of a product rule, which states that $(ab)^n = a^n b^n$. Therefore, we get:

$\left( -3b^4 \right)^2 = (-3)^2 \cdot (b^4)^2 = 9b^8$

Step 2: Simplify the denominator

In the denominator, $-12b^{-9}$ is already written out. We can express this as:

$-12b^{-9}$

Step 3: Put it all together

Now we have the expression:

$\frac{9b^8}{-12b^{-9}}$

Step 4: Simplify the fraction

First, we simplify the constants:

$\frac{9}{-12} = -\frac{3}{4}$

Next, apply the quotient rule for exponents, which states that $\frac{b^m}{b^n} = b^{m-n}$. Here, we have:

$\frac{b^8}{b^{-9}} = b^{8 - (-9)} = b^{8 + 9} = b^{17}$

Final simplified expression:

Putting it all together, we get:

$-\frac{3b^{17}}{4}$

This is the expression in simplest form, with positive exponents.

Would you like further clarification or details on any of the steps? Here are some related questions you might find interesting:

1. How do you handle negative exponents in expressions?
2. Can you explain the quotient rule for exponents in more detail?
3. How do you simplify expressions with both positive and negative exponents?
4. What happens when we divide two terms with the same base and different exponents?
5. How do you deal with fractional coefficients when simplifying algebraic expressions?

Tip: When simplifying exponents, always remember to apply the exponent rules (product, quotient, power of a power) and watch for negative exponents, which can be rewritten as fractions.