Rewrite the denominator using exponent rules.

The denominator is $4^b - 1$. We can express 4 as $2^2$.

$$4^b - 1 = (2^2)^b - 1$$

Using the exponent rule $(x^m)^n = x^{mn}$, we get:

$$(2^2)^b - 1 = 2^{2b} - 1$$

This can also be written as $(2^b)^2 - 1$.

The difference of squares formula is $a^2 - c^2 = (a - c)(a + c)$.
Here, $a = 2^b$ and $c = 1$.

$$(2^b)^2 - 1^2 = (2^b - 1)(2^b + 1)$$

Assuming $2^b - 1 \neq 0$, we can cancel $(2^b - 1)$ from both the numerator and the denominator.

$$\frac{1}{2^b + 1}$$