The image shows the mathematical expression:

$\frac{4b^0 \cdot b^1 \cdot (b^3)^2}{(4b)^2 \cdot (b^{-2})^3 \cdot b^4}$

The instruction is to simplify this expression as much as possible. Let's proceed step-by-step.

1. First, simplify the powers on both the numerator and the denominator.

In the numerator: $4b^0 \cdot b^1 \cdot (b^3)^2 = 4 \cdot 1 \cdot b^1 \cdot b^6 = 4b^7$ (since $b^0 = 1$ and $(b^3)^2 = b^6$).

In the denominator: $(4b)^2 \cdot (b^{-2})^3 \cdot b^4 = 4^2 \cdot b^2 \cdot b^{-6} \cdot b^4 = 16b^{0} = 16$ (since $(4b)^2 = 4^2 \cdot b^2$, and $(b^{-2})^3 = b^{-6}$, and the powers of $b$ add up to $b^2 + (-6) + 4 = b^0 = 1$).

2. Now the expression is simplified to: $\frac{4b^7}{16} = \frac{b^7}{4}$

Thus, the simplified expression is:

$\boxed{\frac{b^7}{4}}$

Would you like a deeper explanation on any step or have more questions? Here are some related questions for further exploration:

1. How do you simplify powers of powers in exponents?
2. What is the rule for multiplying terms with the same base?
3. How do negative exponents work when simplifying expressions?
4. Why does any base raised to the power of zero equal 1?
5. What is the process of combining constants in fractions?

Tip: When dealing with exponents, remember that powers of the same base can be added or subtracted depending on whether you're multiplying or dividing!