We are asked to simplify the expression $\frac{(a^0b^{-2})^2}{b \cdot a b^0}$.

1. Problem Description

We are asked to simplify the expression $\frac{(a^0b^{-2})^2}{b \cdot a b^0}$.

2. Solution Steps

We simplify the given expression using the properties of exponents.

$a^0 = 1$ for $a \neq 0$

$b^0 = 1$ for $b \neq 0$

$(x^m)^n = x^{mn}$

$x^m x^n = x^{m+n}$

$\frac{x^m}{x^n} = x^{m-n}$

First, simplify the numerator:

$(a^0b^{-2})^2 = (1 \cdot b^{-2})^2 = (b^{-2})^2 = b^{-4}$

Next, simplify the denominator:

$b \cdot a \cdot b^0 = b \cdot a \cdot 1 = a \cdot b$

Now, the expression becomes:

$\frac{b^{-4}}{ab} = \frac{b^{-4}}{a^1b^1} = \frac{1}{a} \cdot \frac{b^{-4}}{b^1} = \frac{1}{a} \cdot b^{-4-1} = \frac{1}{a} \cdot b^{-5} = \frac{1}{a} \cdot \frac{1}{b^5} = \frac{1}{ab^5}$

3. Final Answer

$\frac{1}{ab^5}$