The problem asks to simplify the expression $(zx^2y^3)^2 \cdot (xz^2)^2$.

1. Problem Description

The problem asks to simplify the expression $(zx^2y^3)^2 \cdot (xz^2)^2$.

2. Solution Steps

First, we apply the power of a product rule, which states that $(ab)^n = a^n b^n$.

Applying this rule to the first term:

$(zx^2y^3)^2 = z^2(x^2)^2(y^3)^2$

Applying this rule to the second term:

$(xz^2)^2 = x^2(z^2)^2$

Now we use the power of a power rule, which states that $(a^m)^n = a^{m \cdot n}$.

$(x^2)^2 = x^{2 \cdot 2} = x^4$

$(y^3)^2 = y^{3 \cdot 2} = y^6$

$(z^2)^2 = z^{2 \cdot 2} = z^4$

So, our expression becomes:

$z^2x^4y^6 \cdot x^2z^4$

Now we multiply the terms and apply the product of powers rule, which states that $a^m \cdot a^n = a^{m+n}$.

$z^2x^4y^6 \cdot x^2z^4 = x^4x^2y^6z^2z^4 = x^{4+2}y^6z^{2+4} = x^6y^6z^6$

3. Final Answer

The simplified expression is $x^6y^6z^6$.